Subgroups — C₆₇×#32924 #30712175

Order 17152.

1468 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×H(5,𝔽₂)×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3040	C₂	1	2	yes	maximal, complemented
3	C₂×#455×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#453×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#2544	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1887	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1885	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1883	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1882	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1806	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1800	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#137×C₆₇	C₂²	1	2	yes	—
14	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	2	yes	—
15	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic, complemented
18	#520×C₆₇	—	2	2	no	—
19	#520×C₆₇	C₂²	1	2	yes	—
20	#518×C₆₇	—	2	2	no	—
21	#518×C₆₇	—	2	2	no	—
22	#518×C₆₇	C₂²	1	1	yes	characteristic
23	C₄×#41×C₆₇	—	2	2	no	—
24	C₄×#41×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄×D₄×C₆₇	—	2	2	no	—
26	C₂×C₄×D₄×C₆₇	—	2	2	no	—
27	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
28	#464×C₆₇	—	2	2	no	—
29	#464×C₆₇	—	2	2	no	—
30	#463×C₆₇	—	2	2	no	—
31	#463×C₆₇	—	2	2	no	—
32	#455×C₆₇	—	2	4	no	—
33	#453×C₆₇	—	2	4	no	—
34	#423×C₆₇	C₂²	1	2	yes	complemented
35	C₂×#104×C₆₇	C₂²	1	2	yes	—
36	C₂×#104×C₆₇	C₂²	1	2	yes	—
37	#419×C₆₇	C₂²	1	2	yes	complemented
38	#418×C₆₇	C₂²	1	2	yes	complemented
39	C₂×#100×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×#100×C₆₇	C₂²	1	1	yes	characteristic
41	#394×C₆₇	—	2	2	no	—
42	#391×C₆₇	C₂²	1	1	yes	characteristic, complemented
43	#390×C₆₇	C₂²	1	1	yes	characteristic, complemented
44	#386×C₆₇	—	2	2	no	—
45	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
46	#343×C₆₇	C₂²	1	2	yes	complemented
47	#339×C₆₇	C₂²	1	2	yes	complemented
48	#339×C₆₇	C₂²	1	2	yes	complemented
49	H(5,𝔽₂)×C₆₇	—	2	4	no	—
50	H(5,𝔽₂)×C₆₇	—	2	2	no	—
51	H(5,𝔽₂)×C₆₇	D₄	1	4	yes	—
52	C₂×#41×C₆₇	—	2	2	no	—
53	C₂×#41×C₆₇	C₂³	1	2	yes	—
54	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
55	C₂²×D₄×C₆₇	—	2	2	no	—
56	C₂²×D₄×C₆₇	—	2	2	no	—
57	C₂²×D₄×C₆₇	—	2	2	no	—
58	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
59	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₆₇	—	2	2	no	—
61	C₂³×C₄×C₆₇	D₄	1	2	yes	—
62	#137×C₆₇	—	2	4	no	—
63	#137×C₆₇	—	2	4	no	—
64	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
65	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
66	C₂×Q₁₆×C₆₇	—	2	2	no	—
67	C₂×SD₁₆×C₆₇	—	2	2	no	—
68	C₂×SD₁₆×C₆₇	—	2	2	no	—
69	C₂×D₈×C₆₇	—	2	2	no	—
70	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
71	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
72	#127×C₆₇	—	2	2	no	—
73	#127×C₆₇	—	2	2	no	—
74	#124×C₆₇	—	2	2	no	—
75	#124×C₆₇	—	2	2	no	—
76	#124×C₆₇	—	2	2	no	—
77	#124×C₆₇	—	2	2	no	—
78	C₄×Q₈×C₆₇	—	2	2	no	—
79	C₄×Q₈×C₆₇	D₄	1	2	yes	complemented
80	C₄×Q₈×C₆₇	D₄	1	2	yes	complemented
81	C₄×D₄×C₆₇	—	2	4	no	—
82	C₄×D₄×C₆₇	—	2	4	no	—
83	C₄×D₄×C₆₇	—	2	4	no	—
84	C₄×D₄×C₆₇	—	2	4	no	—
85	C₄×D₄×C₆₇	—	2	2	no	—
86	C₄×D₄×C₆₇	—	2	2	no	—
87	C₄×D₄×C₆₇	—	2	4	no	—
88	C₄×D₄×C₆₇	—	2	2	no	—
89	C₄×D₄×C₆₇	—	2	2	no	—
90	C₄×D₄×C₆₇	—	2	2	no	—
91	C₄×D₄×C₆₇	—	2	2	no	—
92	C₄×D₄×C₆₇	—	2	2	no	—
93	C₄×D₄×C₆₇	D₄	1	2	yes	complemented
94	C₄×D₄×C₆₇	D₄	1	2	yes	complemented
95	#117×C₆₇	—	2	4	no	—
96	#117×C₆₇	—	2	2	no	—
97	#117×C₆₇	—	2	2	no	—
98	#117×C₆₇	C₂³	1	2	yes	—
99	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
100	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
101	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
102	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
103	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
104	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
105	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
106	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
107	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
108	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
109	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
110	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
111	C₂×C₄²×C₆₇	—	2	2	no	—
112	C₂×C₄²×C₆₇	—	2	2	no	—
113	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
114	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
116	C₄⋊C₈×C₆₇	—	2	2	no	—
117	C₄⋊C₈×C₆₇	—	2	2	no	—
118	#104×C₆₇	—	2	4	no	—
119	#104×C₆₇	—	2	4	no	—
120	#103×C₆₇	—	2	2	no	—
121	#103×C₆₇	—	2	2	no	—
122	#102×C₆₇	—	2	2	no	—
123	#102×C₆₇	—	2	2	no	—
124	#100×C₆₇	—	2	2	no	—
125	#100×C₆₇	—	2	2	no	—
126	#95×C₆₇	—	2	1	no	—
127	#95×C₆₇	—	2	2	no	—
128	C₂⁴×C₆₇	—	2	2	no	—
129	C₂⁴×C₆₇	—	2	1	no	—
130	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
131	#41×C₆₇	—	2	8	no	—
132	#41×C₆₇	—	2	8	no	—
133	#41×C₆₇	—	2	2	no	—
134	#41×C₆₇	—	2	2	no	—
135	#41×C₆₇	C₂×D₄	1	4	yes	—
136	#41×C₆₇	#41	1	4	yes	—
137	C₂×Q₈×C₆₇	—	2	2	no	—
138	C₂×Q₈×C₆₇	—	2	1	no	—
139	C₂×Q₈×C₆₇	—	2	1	no	—
140	C₂×Q₈×C₆₇	—	2	2	no	—
141	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
142	C₂×D₄×C₆₇	—	2	4	no	—
143	C₂×D₄×C₆₇	—	2	8	no	—
144	C₂×D₄×C₆₇	—	2	8	no	—
145	C₂×D₄×C₆₇	—	2	8	no	—
146	C₂×D₄×C₆₇	—	2	2	no	—
147	C₂×D₄×C₆₇	—	2	8	no	—
148	C₂×D₄×C₆₇	—	2	2	no	—
149	C₂×D₄×C₆₇	—	2	2	no	—
150	C₂×D₄×C₆₇	—	2	1	no	—
151	C₂×D₄×C₆₇	—	2	2	no	—
152	C₂×D₄×C₆₇	—	2	1	no	—
153	C₂×D₄×C₆₇	—	2	2	no	—
154	C₂×D₄×C₆₇	—	4	4	no	—
155	C₂×D₄×C₆₇	—	4	4	no	—
156	C₂×D₄×C₆₇	—	8	2	no	—
157	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
158	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
159	C₂×D₄×C₆₇	#41	1	2	yes	—
160	C₂²×C₄×C₆₇	—	2	2	no	—
161	C₂²×C₄×C₆₇	—	2	2	no	—
162	C₂²×C₄×C₆₇	—	2	1	no	—
163	C₂²×C₄×C₆₇	—	2	2	no	—
164	C₂²×C₄×C₆₇	—	2	4	no	—
165	C₂²×C₄×C₆₇	—	2	2	no	—
166	C₂²×C₄×C₆₇	—	2	2	no	—
167	C₂²×C₄×C₆₇	—	2	2	no	—
168	C₂²×C₄×C₆₇	—	2	2	no	—
169	C₂²×C₄×C₆₇	—	2	2	no	—
170	C₂²×C₄×C₆₇	—	2	1	no	—
171	C₂²×C₄×C₆₇	—	2	1	no	—
172	C₂²×C₄×C₆₇	—	8	2	no	—
173	C₂²×C₄×C₆₇	—	8	4	no	—
174	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
175	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
178	Q₁₆×C₆₇	—	4	4	no	—
179	SD₁₆×C₆₇	—	4	4	no	—
180	SD₁₆×C₆₇	—	4	4	no	—
181	D₈×C₆₇	—	4	4	no	—
182	M₁₆×C₆₇	—	2	2	no	—
183	M₁₆×C₆₇	—	2	2	no	—
184	M₁₆×C₆₇	—	2	2	no	—
185	M₁₆×C₆₇	—	2	2	no	—
186	C₂×C₈×C₆₇	—	2	2	no	—
187	C₂×C₈×C₆₇	—	2	2	no	—
188	C₄⋊C₄×C₆₇	—	2	4	no	—
189	C₄⋊C₄×C₆₇	—	2	2	no	—
190	C₄⋊C₄×C₆₇	—	2	2	no	—
191	C₄⋊C₄×C₆₇	—	2	2	no	—
192	C₄⋊C₄×C₆₇	—	4	2	no	—
193	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
194	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
195	C₂²⋊C₄×C₆₇	—	2	4	no	—
196	C₂²⋊C₄×C₆₇	—	2	4	no	—
197	C₂²⋊C₄×C₆₇	—	2	2	no	—
198	C₂²⋊C₄×C₆₇	—	2	2	no	—
199	C₂²⋊C₄×C₆₇	—	2	2	no	—
200	C₂²⋊C₄×C₆₇	—	2	4	no	—
201	C₂²⋊C₄×C₆₇	—	4	4	no	—
202	C₂²⋊C₄×C₆₇	—	4	1	no	—
203	C₂²⋊C₄×C₆₇	—	4	2	no	—
204	C₂²⋊C₄×C₆₇	—	4	1	no	—
205	C₂²⋊C₄×C₆₇	—	8	2	no	—
206	C₄²×C₆₇	—	2	4	no	—
207	C₄²×C₆₇	—	2	2	no	—
208	C₄²×C₆₇	—	2	2	no	—
209	C₄²×C₆₇	—	2	2	no	—
210	C₄²×C₆₇	—	2	1	no	—
211	C₄²×C₆₇	—	2	1	no	—
212	C₄²×C₆₇	—	2	1	no	—
213	C₄²×C₆₇	—	2	1	no	—
214	C₄²×C₆₇	C₂×D₄	1	2	yes	—
215	C₄²×C₆₇	C₂×D₄	1	2	yes	—
216	C₂³×C₆₇	—	2	8	no	—
217	C₂³×C₆₇	—	2	2	no	—
218	C₂³×C₆₇	—	2	2	no	—
219	C₂³×C₆₇	—	2	2	no	—
220	C₂³×C₆₇	—	4	2	no	—
221	C₂³×C₆₇	—	4	2	no	—
222	C₂³×C₆₇	—	4	2	no	—
223	C₂³×C₆₇	—	4	2	no	—
224	C₂³×C₆₇	—	4	1	no	—
225	C₂³×C₆₇	—	4	1	no	—
226	C₂³×C₆₇	—	4	1	no	—
227	C₂³×C₆₇	—	8	2	no	—
228	C₂³×C₆₇	—	8	2	no	—
229	C₂³×C₆₇	#121	1	4	yes	—
230	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
231	Q₈×C₆₇	—	2	2	no	—
232	Q₈×C₆₇	—	4	2	no	—
233	Q₈×C₆₇	—	4	2	no	—
234	Q₈×C₆₇	#121	1	4	yes	—
235	D₄×C₆₇	—	2	16	no	—
236	D₄×C₆₇	—	2	4	no	—
237	D₄×C₆₇	—	2	4	no	—
238	D₄×C₆₇	—	2	4	no	—
239	D₄×C₆₇	—	2	2	no	—
240	D₄×C₆₇	—	2	4	no	—
241	D₄×C₆₇	—	4	2	no	—
242	D₄×C₆₇	—	4	2	no	—
243	D₄×C₆₇	—	8	8	no	—
244	D₄×C₆₇	#121	1	4	yes	—
245	C₂×C₄×C₆₇	—	2	8	no	—
246	C₂×C₄×C₆₇	—	2	2	no	—
247	C₂×C₄×C₆₇	—	2	4	no	—
248	C₂×C₄×C₆₇	—	2	4	no	—
249	C₂×C₄×C₆₇	—	2	2	no	—
250	C₂×C₄×C₆₇	—	2	2	no	—
251	C₂×C₄×C₆₇	—	2	2	no	—
252	C₂×C₄×C₆₇	—	2	2	no	—
253	C₂×C₄×C₆₇	—	4	1	no	—
254	C₂×C₄×C₆₇	—	4	1	no	—
255	C₂×C₄×C₆₇	—	4	4	no	—
256	C₂×C₄×C₆₇	—	4	4	no	—
257	C₂×C₄×C₆₇	—	4	1	no	—
258	C₂×C₄×C₆₇	—	4	1	no	—
259	C₂×C₄×C₆₇	—	4	2	no	—
260	C₂×C₄×C₆₇	—	4	1	no	—
261	C₂×C₄×C₆₇	—	4	1	no	—
262	C₂×C₄×C₆₇	—	4	1	no	—
263	C₂×C₄×C₆₇	—	4	1	no	—
264	C₂×C₄×C₆₇	—	4	1	no	—
265	C₂×C₄×C₆₇	—	4	1	no	—
266	C₂×C₄×C₆₇	—	8	2	no	—
267	C₂×C₄×C₆₇	—	8	2	no	—
268	C₂×C₄×C₆₇	—	8	2	no	—
269	C₂×C₄×C₆₇	—	8	4	no	—
270	C₂×C₄×C₆₇	—	8	2	no	—
271	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
272	C₂×C₄×C₆₇	#120	1	2	yes	—
273	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
274	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
275	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
276	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
277	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
278	C₈×C₆₇	—	4	2	no	—
279	C₈×C₆₇	—	4	2	no	—
280	C₂²×C₆₇	—	2	1	no	—
281	C₂²×C₆₇	—	2	1	no	—
282	C₂²×C₆₇	—	2	4	no	—
283	C₂²×C₆₇	—	2	4	no	—
284	C₂²×C₆₇	—	4	4	no	—
285	C₂²×C₆₇	—	4	2	no	—
286	C₂²×C₆₇	—	4	2	no	—
287	C₂²×C₆₇	—	4	2	no	—
288	C₂²×C₆₇	—	4	2	no	—
289	C₂²×C₆₇	—	8	2	no	—
290	C₂²×C₆₇	—	8	2	no	—
291	C₂²×C₆₇	—	8	2	no	—
292	C₂²×C₆₇	—	8	8	no	—
293	C₂²×C₆₇	—	8	1	no	—
294	C₂²×C₆₇	—	8	1	no	—
295	C₂²×C₆₇	—	8	1	no	—
296	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
297	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
298	C₂²×C₆₇	#525	1	1	yes	characteristic
299	C₄×C₆₇	—	2	4	no	—
300	C₄×C₆₇	—	2	2	no	—
301	C₄×C₆₇	—	2	1	no	—
302	C₄×C₆₇	—	2	1	no	—
303	C₄×C₆₇	—	4	2	no	—
304	C₄×C₆₇	—	4	2	no	—
305	C₄×C₆₇	—	4	2	no	—
306	C₄×C₆₇	—	4	2	no	—
307	C₄×C₆₇	—	8	1	no	—
308	C₄×C₆₇	—	8	1	no	—
309	C₄×C₆₇	C₂×#121	1	2	yes	—
310	C₄×C₆₇	#525	1	2	yes	—
311	#32924	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
312	C₂×C₆₇	—	2	1	no	—
313	C₂×C₆₇	—	2	1	no	—
314	C₂×C₆₇	—	4	4	no	—
315	C₂×C₆₇	—	4	4	no	—
316	C₂×C₆₇	—	16	1	no	—
317	C₂×C₆₇	#2847	1	1	yes	characteristic
318	C₂×C₆₇	#3018	1	1	yes	characteristic
319	C₂×C₆₇	#3019	1	1	yes	characteristic
320	C₄×H(5,𝔽₂)	C₂×C₆₇	1	1	yes	characteristic, complemented
321	#3040	C₂×C₆₇	1	2	yes	complemented
322	C₂×#455	C₂×C₆₇	1	1	yes	characteristic, complemented
323	C₂×#453	C₂×C₆₇	1	1	yes	characteristic
324	#2544	C₂×C₆₇	1	1	yes	characteristic, complemented
325	#1887	C₂×C₆₇	1	1	yes	characteristic, complemented
326	#1885	C₂×C₆₇	1	1	yes	characteristic, complemented
327	#1883	C₂×C₆₇	1	2	yes	complemented
328	#1882	C₂×C₆₇	1	2	yes	complemented
329	#1806	C₂×C₆₇	1	2	yes	complemented
330	#1800	C₂×C₆₇	1	1	yes	characteristic, complemented
331	C₆₇	#32924	1	1	yes	characteristic, direct product factor, Sylow
332	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	1	yes	characteristic
333	C₂×#137	C₂²×C₆₇	1	2	yes	—
334	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	2	yes	—
335	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
336	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
337	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic, complemented
338	#520	—	2	2	no	—
339	#520	C₂²×C₆₇	1	2	yes	—
340	#518	—	2	2	no	—
341	#518	—	2	2	no	—
342	#518	C₂²×C₆₇	1	1	yes	characteristic
343	C₄×#41	—	2	2	no	—
344	C₄×#41	C₂²×C₆₇	1	2	yes	—
345	C₂×C₄×D₄	—	2	2	no	—
346	C₂×C₄×D₄	—	2	2	no	—
347	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
348	#464	—	2	2	no	—
349	#464	—	2	2	no	—
350	#463	—	2	2	no	—
351	#463	—	2	2	no	—
352	#455	—	2	4	no	—
353	#453	—	2	4	no	—
354	#423	C₂²×C₆₇	1	2	yes	complemented
355	C₂×#104	C₂²×C₆₇	1	2	yes	—
356	C₂×#104	C₂²×C₆₇	1	2	yes	—
357	#419	C₂²×C₆₇	1	2	yes	complemented
358	#418	C₂²×C₆₇	1	2	yes	complemented
359	C₂×#100	C₂²×C₆₇	1	1	yes	characteristic
360	C₂×#100	C₂²×C₆₇	1	1	yes	characteristic
361	#394	—	2	2	no	—
362	#391	C₂²×C₆₇	1	1	yes	characteristic, complemented
363	#390	C₂²×C₆₇	1	1	yes	characteristic, complemented
364	#386	—	2	2	no	—
365	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
366	#343	C₂²×C₆₇	1	2	yes	complemented
367	#339	C₂²×C₆₇	1	2	yes	complemented
368	#339	C₂²×C₆₇	1	2	yes	complemented
369	H(5,𝔽₂)	—	2	4	no	—
370	H(5,𝔽₂)	—	2	2	no	—
371	H(5,𝔽₂)	D₄×C₆₇	1	4	yes	—
372	C₂×#41	—	2	2	no	—
373	C₂×#41	C₂³×C₆₇	1	2	yes	—
374	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
375	C₂²×D₄	—	2	2	no	—
376	C₂²×D₄	—	2	2	no	—
377	C₂²×D₄	—	2	2	no	—
378	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
379	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
380	C₂³×C₄	—	2	2	no	—
381	C₂³×C₄	D₄×C₆₇	1	2	yes	—
382	#137	—	2	4	no	—
383	#137	—	2	4	no	—
384	AGL(1,ℤ/8)	—	2	4	no	—
385	AGL(1,ℤ/8)	—	2	4	no	—
386	C₂×Q₁₆	—	2	2	no	—
387	C₂×SD₁₆	—	2	2	no	—
388	C₂×SD₁₆	—	2	2	no	—
389	C₂×D₈	—	2	2	no	—
390	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
391	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
392	#127	—	2	2	no	—
393	#127	—	2	2	no	—
394	#124	—	2	2	no	—
395	#124	—	2	2	no	—
396	#124	—	2	2	no	—
397	#124	—	2	2	no	—
398	C₄×Q₈	—	2	2	no	—
399	C₄×Q₈	D₄×C₆₇	1	2	yes	complemented
400	C₄×Q₈	D₄×C₆₇	1	2	yes	complemented
401	C₄×D₄	—	2	4	no	—
402	C₄×D₄	—	2	4	no	—
403	C₄×D₄	—	2	4	no	—
404	C₄×D₄	—	2	4	no	—
405	C₄×D₄	—	2	2	no	—
406	C₄×D₄	—	2	2	no	—
407	C₄×D₄	—	2	4	no	—
408	C₄×D₄	—	2	2	no	—
409	C₄×D₄	—	2	2	no	—
410	C₄×D₄	—	2	2	no	—
411	C₄×D₄	—	2	2	no	—
412	C₄×D₄	—	2	2	no	—
413	C₄×D₄	D₄×C₆₇	1	2	yes	complemented
414	C₄×D₄	D₄×C₆₇	1	2	yes	complemented
415	#117	—	2	4	no	—
416	#117	—	2	2	no	—
417	#117	—	2	2	no	—
418	#117	C₂³×C₆₇	1	2	yes	—
419	C₂×C₄⋊C₄	—	2	2	no	—
420	C₂×C₄⋊C₄	—	2	2	no	—
421	C₂×C₄⋊C₄	—	2	1	no	—
422	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
423	C₂×C₂²⋊C₄	—	2	2	no	—
424	C₂×C₂²⋊C₄	—	2	4	no	—
425	C₂×C₂²⋊C₄	—	2	2	no	—
426	C₂×C₂²⋊C₄	—	2	2	no	—
427	C₂×C₂²⋊C₄	—	2	1	no	—
428	C₂×C₂²⋊C₄	—	2	1	no	—
429	C₂×C₂²⋊C₄	—	2	2	no	—
430	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
431	C₂×C₄²	—	2	2	no	—
432	C₂×C₄²	—	2	2	no	—
433	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
434	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
435	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
436	C₄⋊C₈	—	2	2	no	—
437	C₄⋊C₈	—	2	2	no	—
438	#104	—	2	4	no	—
439	#104	—	2	4	no	—
440	#103	—	2	2	no	—
441	#103	—	2	2	no	—
442	#102	—	2	2	no	—
443	#102	—	2	2	no	—
444	#100	—	2	2	no	—
445	#100	—	2	2	no	—
446	#95	—	2	1	no	—
447	#95	—	2	2	no	—
448	C₂⁴	—	2	2	no	—
449	C₂⁴	—	2	1	no	—
450	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
451	#41	—	2	8	no	—
452	#41	—	2	8	no	—
453	#41	—	2	2	no	—
454	#41	—	2	2	no	—
455	#41	C₂×D₄×C₆₇	1	4	yes	—
456	#41	#41×C₆₇	1	4	yes	—
457	C₂×Q₈	—	2	2	no	—
458	C₂×Q₈	—	2	1	no	—
459	C₂×Q₈	—	2	1	no	—
460	C₂×Q₈	—	2	2	no	—
461	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
462	C₂×D₄	—	2	4	no	—
463	C₂×D₄	—	2	8	no	—
464	C₂×D₄	—	2	8	no	—
465	C₂×D₄	—	2	8	no	—
466	C₂×D₄	—	2	2	no	—
467	C₂×D₄	—	2	8	no	—
468	C₂×D₄	—	2	2	no	—
469	C₂×D₄	—	2	2	no	—
470	C₂×D₄	—	2	1	no	—
471	C₂×D₄	—	2	2	no	—
472	C₂×D₄	—	2	1	no	—
473	C₂×D₄	—	2	2	no	—
474	C₂×D₄	—	4	4	no	—
475	C₂×D₄	—	4	4	no	—
476	C₂×D₄	—	8	2	no	—
477	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
478	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
479	C₂×D₄	#41×C₆₇	1	2	yes	—
480	C₂²×C₄	—	2	2	no	—
481	C₂²×C₄	—	2	2	no	—
482	C₂²×C₄	—	2	1	no	—
483	C₂²×C₄	—	2	2	no	—
484	C₂²×C₄	—	2	4	no	—
485	C₂²×C₄	—	2	2	no	—
486	C₂²×C₄	—	2	2	no	—
487	C₂²×C₄	—	2	2	no	—
488	C₂²×C₄	—	2	2	no	—
489	C₂²×C₄	—	2	2	no	—
490	C₂²×C₄	—	2	1	no	—
491	C₂²×C₄	—	2	1	no	—
492	C₂²×C₄	—	8	2	no	—
493	C₂²×C₄	—	8	4	no	—
494	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
495	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
496	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
497	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
498	Q₁₆	—	4	4	no	—
499	SD₁₆	—	4	4	no	—
500	SD₁₆	—	4	4	no	—
501	D₈	—	4	4	no	—
502	M₁₆	—	2	2	no	—
503	M₁₆	—	2	2	no	—
504	M₁₆	—	2	2	no	—
505	M₁₆	—	2	2	no	—
506	C₂×C₈	—	2	2	no	—
507	C₂×C₈	—	2	2	no	—
508	C₄⋊C₄	—	2	4	no	—
509	C₄⋊C₄	—	2	2	no	—
510	C₄⋊C₄	—	2	2	no	—
511	C₄⋊C₄	—	2	2	no	—
512	C₄⋊C₄	—	4	2	no	—
513	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
514	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
515	C₂²⋊C₄	—	2	4	no	—
516	C₂²⋊C₄	—	2	4	no	—
517	C₂²⋊C₄	—	2	2	no	—
518	C₂²⋊C₄	—	2	2	no	—
519	C₂²⋊C₄	—	2	2	no	—
520	C₂²⋊C₄	—	2	4	no	—
521	C₂²⋊C₄	—	4	4	no	—
522	C₂²⋊C₄	—	4	1	no	—
523	C₂²⋊C₄	—	4	2	no	—
524	C₂²⋊C₄	—	4	1	no	—
525	C₂²⋊C₄	—	8	2	no	—
526	C₄²	—	2	4	no	—
527	C₄²	—	2	2	no	—
528	C₄²	—	2	2	no	—
529	C₄²	—	2	2	no	—
530	C₄²	—	2	1	no	—
531	C₄²	—	2	1	no	—
532	C₄²	—	2	1	no	—
533	C₄²	—	2	1	no	—
534	C₄²	C₂×D₄×C₆₇	1	2	yes	—
535	C₄²	C₂×D₄×C₆₇	1	2	yes	—
536	C₂³	—	2	8	no	—
537	C₂³	—	2	2	no	—
538	C₂³	—	2	2	no	—
539	C₂³	—	2	2	no	—
540	C₂³	—	4	2	no	—
541	C₂³	—	4	2	no	—
542	C₂³	—	4	2	no	—
543	C₂³	—	4	2	no	—
544	C₂³	—	4	1	no	—
545	C₂³	—	4	1	no	—
546	C₂³	—	4	1	no	—
547	C₂³	—	8	2	no	—
548	C₂³	—	8	2	no	—
549	C₂³	#121×C₆₇	1	4	yes	—
550	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
551	Q₈	—	2	2	no	—
552	Q₈	—	4	2	no	—
553	Q₈	—	4	2	no	—
554	Q₈	#121×C₆₇	1	4	yes	—
555	D₄	—	2	16	no	—
556	D₄	—	2	4	no	—
557	D₄	—	2	4	no	—
558	D₄	—	2	4	no	—
559	D₄	—	2	2	no	—
560	D₄	—	2	4	no	—
561	D₄	—	4	2	no	—
562	D₄	—	4	2	no	—
563	D₄	—	8	8	no	—
564	D₄	#121×C₆₇	1	4	yes	—
565	C₂×C₄	—	2	8	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	4	no	—
568	C₂×C₄	—	2	4	no	—
569	C₂×C₄	—	2	2	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	4	1	no	—
574	C₂×C₄	—	4	1	no	—
575	C₂×C₄	—	4	4	no	—
576	C₂×C₄	—	4	4	no	—
577	C₂×C₄	—	4	1	no	—
578	C₂×C₄	—	4	1	no	—
579	C₂×C₄	—	4	2	no	—
580	C₂×C₄	—	4	1	no	—
581	C₂×C₄	—	4	1	no	—
582	C₂×C₄	—	4	1	no	—
583	C₂×C₄	—	4	1	no	—
584	C₂×C₄	—	4	1	no	—
585	C₂×C₄	—	4	1	no	—
586	C₂×C₄	—	8	2	no	—
587	C₂×C₄	—	8	2	no	—
588	C₂×C₄	—	8	4	no	—
589	C₂×C₄	—	8	2	no	—
590	C₂×C₄	—	8	2	no	—
591	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
592	C₂×C₄	#120×C₆₇	1	2	yes	—
593	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
594	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
595	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
596	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
597	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
598	C₈	—	4	2	no	—
599	C₈	—	4	2	no	—
600	C₂²	—	2	1	no	—
601	C₂²	—	2	1	no	—
602	C₂²	—	2	4	no	—
603	C₂²	—	2	4	no	—
604	C₂²	—	4	4	no	—
605	C₂²	—	4	2	no	—
606	C₂²	—	4	2	no	—
607	C₂²	—	4	2	no	—
608	C₂²	—	4	2	no	—
609	C₂²	—	8	2	no	—
610	C₂²	—	8	2	no	—
611	C₂²	—	8	2	no	—
612	C₂²	—	8	8	no	—
613	C₂²	—	8	1	no	—
614	C₂²	—	8	1	no	—
615	C₂²	—	8	1	no	—
616	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
617	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
618	C₂²	#525×C₆₇	1	1	yes	characteristic
619	C₄	—	2	2	no	—
620	C₄	—	2	1	no	—
621	C₄	—	2	1	no	—
622	C₄	—	2	4	no	—
623	C₄	—	4	2	no	—
624	C₄	—	4	2	no	—
625	C₄	—	4	2	no	—
626	C₄	—	4	2	no	—
627	C₄	—	8	1	no	—
628	C₄	—	8	1	no	—
629	C₄	C₂×#121×C₆₇	1	2	yes	—
630	C₄	#525×C₆₇	1	2	yes	—
631	C₂	—	2	1	no	—
632	C₂	—	2	1	no	—
633	C₂	—	4	4	no	—
634	C₂	—	4	4	no	—
635	C₂	—	16	1	no	—
636	C₂	C₆₇×#2847	1	1	yes	characteristic
637	C₂	C₆₇×#3018	1	1	yes	characteristic
638	C₂	C₆₇×#3019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#453×C₆₇	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₄×H(5,𝔽₂)×C₆₇	C₂	semidirect
2. C₆₇×#3040	C₂	semidirect
3. C₂×#455×C₆₇	C₂	semidirect
5. C₆₇×#2544	C₂	semidirect
6. C₆₇×#1887	C₂	semidirect
7. C₆₇×#1885	C₂	semidirect
8. C₆₇×#1883	C₂	semidirect
9. C₆₇×#1882	C₂	semidirect
10. C₆₇×#1806	C₂	semidirect
11. C₆₇×#1800	C₂	semidirect
17. C₂×#124×C₆₇	C₂²	semidirect
34. #423×C₆₇	C₂²	semidirect
37. #419×C₆₇	C₂²	semidirect
38. #418×C₆₇	C₂²	semidirect
42. #391×C₆₇	C₂²	semidirect
43. #390×C₆₇	C₂²	semidirect
45. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
46. #343×C₆₇	C₂²	semidirect
47. #339×C₆₇	C₂²	semidirect
48. #339×C₆₇	C₂²	semidirect
79. C₄×Q₈×C₆₇	D₄	semidirect
80. C₄×Q₈×C₆₇	D₄	semidirect
93. C₄×D₄×C₆₇	D₄	semidirect
94. C₄×D₄×C₆₇	D₄	semidirect
311. #32924	C₆₇	direct (N × H)
320. C₄×H(5,𝔽₂)	C₂×C₆₇	semidirect
321. #3040	C₂×C₆₇	semidirect
322. C₂×#455	C₂×C₆₇	semidirect
324. #2544	C₂×C₆₇	semidirect
325. #1887	C₂×C₆₇	semidirect
326. #1885	C₂×C₆₇	semidirect
327. #1883	C₂×C₆₇	semidirect
328. #1882	C₂×C₆₇	semidirect
329. #1806	C₂×C₆₇	semidirect
330. #1800	C₂×C₆₇	semidirect
331. C₆₇	#32924	direct (N × H)
337. C₂×#124	C₂²×C₆₇	semidirect
354. #423	C₂²×C₆₇	semidirect
357. #419	C₂²×C₆₇	semidirect
358. #418	C₂²×C₆₇	semidirect
362. #391	C₂²×C₆₇	semidirect
363. #390	C₂²×C₆₇	semidirect
365. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
366. #343	C₂²×C₆₇	semidirect
367. #339	C₂²×C₆₇	semidirect
368. #339	C₂²×C₆₇	semidirect
399. C₄×Q₈	D₄×C₆₇	semidirect
400. C₄×Q₈	D₄×C₆₇	semidirect
413. C₄×D₄	D₄×C₆₇	semidirect
414. C₄×D₄	D₄×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×H(5,𝔽₂)×C₆₇	2
C₆₇×#3040	2
C₂×#455×C₆₇	2
C₂×#453×C₆₇	2
C₆₇×#2544	2
C₆₇×#1887	2
C₆₇×#1885	2
C₆₇×#1883	2
C₆₇×#1882	2
C₆₇×#1806	2
C₆₇×#1800	2
#32924	67

Subgroups — C67×#32924 #30712175

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32924 #30712175