Subgroups — C₆₇×#21782 #30701033

Order 17152.

1382 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3498	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2888	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2883	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1911	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1910	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1887	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1884	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1827	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1795	C₂	1	2	yes	maximal, complemented
10	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	1	yes	characteristic
12	#568×C₆₇	C₂²	1	2	yes	complemented
13	#550×C₆₇	—	2	2	no	—
14	D₄²×C₆₇	—	2	4	no	—
15	D₄²×C₆₇	—	2	2	no	—
16	#535×C₆₇	C₄	1	2	yes	complemented
17	#535×C₆₇	C₄	1	2	yes	complemented
18	#535×C₆₇	C₂²	1	1	yes	characteristic
19	#535×C₆₇	C₂²	1	1	yes	characteristic
20	#532×C₆₇	C₄	1	2	yes	complemented
21	#532×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#127×C₆₇	C₄	1	2	yes	complemented
23	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
24	#518×C₆₇	C₂²	1	2	yes	complemented
25	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
26	#421×C₆₇	C₂²	1	4	yes	complemented
27	#418×C₆₇	C₂²	1	4	yes	complemented
28	#413×C₆₇	—	2	2	no	—
29	#413×C₆₇	C₂²	1	2	yes	—
30	C₂×#100×C₆₇	—	2	2	no	—
31	C₂×#100×C₆₇	C₂²	1	2	yes	complemented
32	#410×C₆₇	C₂²	1	2	yes	—
33	#410×C₆₇	C₂²	1	2	yes	complemented
34	C₂×#99×C₆₇	—	2	4	no	—
35	#339×C₆₇	C₂²	1	4	yes	complemented
36	H(3,ℤ/4)×C₆₇	C₂²	1	4	yes	complemented
37	H(5,𝔽₂)×C₆₇	—	4	4	no	—
38	H(5,𝔽₂)×C₆₇	—	4	4	no	—
39	C₂×#41×C₆₇	—	2	1	no	—
40	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
41	C₂×#41×C₆₇	C₂×C₄	1	2	yes	complemented
42	C₂×#41×C₆₇	D₄	1	2	yes	—
43	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
44	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
45	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
46	C₂²×D₄×C₆₇	—	2	4	no	—
47	C₂²×D₄×C₆₇	—	2	2	no	—
48	C₂²×D₄×C₆₇	—	2	2	no	—
49	C₂²×D₄×C₆₇	—	2	1	no	—
50	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
51	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	complemented
52	C₂²×D₄×C₆₇	D₄	1	4	yes	—
53	C₂²×D₄×C₆₇	D₄	1	2	yes	—
54	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
55	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
56	#131×C₆₇	—	4	4	no	—
57	C₂×M₁₆×C₆₇	—	2	4	no	—
58	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
59	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
60	#127×C₆₇	—	2	4	no	—
61	#127×C₆₇	—	2	2	no	—
62	#127×C₆₇	C₂×C₄	1	2	yes	—
63	#127×C₆₇	C₂×C₄	1	2	yes	—
64	#127×C₆₇	C₂×C₄	1	2	yes	—
65	#127×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
66	#127×C₆₇	C₂×C₄	1	2	yes	complemented
67	#124×C₆₇	C₂×C₄	1	2	yes	—
68	#124×C₆₇	C₂×C₄	1	2	yes	complemented
69	#124×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
70	#124×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
71	#121×C₆₇	—	2	4	no	—
72	#121×C₆₇	—	2	4	no	—
73	#121×C₆₇	—	2	2	no	—
74	#121×C₆₇	—	2	2	no	—
75	#121×C₆₇	—	2	4	no	—
76	#121×C₆₇	—	2	2	no	—
77	#120×C₆₇	—	2	4	no	—
78	#120×C₆₇	—	2	2	no	—
79	#120×C₆₇	—	2	4	no	—
80	#120×C₆₇	—	2	2	no	—
81	C₄×Q₈×C₆₇	—	2	1	no	—
82	C₄×D₄×C₆₇	—	2	2	no	—
83	C₄×D₄×C₆₇	—	2	1	no	—
84	C₄×D₄×C₆₇	—	2	4	no	—
85	C₄×D₄×C₆₇	—	2	2	no	—
86	C₄×D₄×C₆₇	—	4	4	no	—
87	#117×C₆₇	C₂×C₄	1	2	yes	complemented
88	#117×C₆₇	C₂³	1	4	yes	—
89	#117×C₆₇	C₂³	1	1	yes	characteristic
90	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
91	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	complemented
92	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
93	#104×C₆₇	—	4	4	no	—
94	#102×C₆₇	—	4	4	no	—
95	#101×C₆₇	—	2	2	no	—
96	#100×C₆₇	—	2	2	no	—
97	#100×C₆₇	—	2	4	no	—
98	#99×C₆₇	—	2	4	no	—
99	#99×C₆₇	—	2	4	no	—
100	C₂⁴×C₆₇	—	2	2	no	—
101	C₂⁴×C₆₇	—	2	1	no	—
102	C₂⁴×C₆₇	C₂²⋊C₄	1	4	yes	—
103	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
104	#41×C₆₇	—	2	4	no	—
105	#41×C₆₇	—	2	4	no	—
106	#41×C₆₇	—	2	4	no	—
107	#41×C₆₇	—	4	4	no	—
108	#41×C₆₇	—	4	4	no	—
109	#41×C₆₇	—	4	4	no	—
110	#41×C₆₇	—	4	4	no	—
111	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₂×D₄×C₆₇	—	2	4	no	—
115	C₂×D₄×C₆₇	—	2	2	no	—
116	C₂×D₄×C₆₇	—	2	2	no	—
117	C₂×D₄×C₆₇	—	2	1	no	—
118	C₂×D₄×C₆₇	—	2	4	no	—
119	C₂×D₄×C₆₇	—	2	4	no	—
120	C₂×D₄×C₆₇	—	2	4	no	—
121	C₂×D₄×C₆₇	—	2	2	no	—
122	C₂×D₄×C₆₇	—	2	1	no	—
123	C₂×D₄×C₆₇	—	2	4	no	—
124	C₂×D₄×C₆₇	—	2	4	no	—
125	C₂×D₄×C₆₇	—	4	4	no	—
126	C₂×D₄×C₆₇	—	4	4	no	—
127	C₂×D₄×C₆₇	—	4	4	no	—
128	C₂×D₄×C₆₇	—	4	8	no	—
129	C₂×D₄×C₆₇	—	4	8	no	—
130	C₂×D₄×C₆₇	—	4	4	no	—
131	C₂×D₄×C₆₇	—	8	4	no	—
132	C₂×D₄×C₆₇	—	8	8	no	—
133	C₂×D₄×C₆₇	C₂²⋊C₄	1	4	yes	—
134	C₂×D₄×C₆₇	C₂²⋊C₄	1	4	yes	—
135	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
136	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
137	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
138	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
139	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
140	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
141	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
142	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
143	C₂²×C₄×C₆₇	—	2	2	no	—
144	C₂²×C₄×C₆₇	—	2	1	no	—
145	C₂²×C₄×C₆₇	—	2	2	no	—
146	C₂²×C₄×C₆₇	—	2	1	no	—
147	C₂²×C₄×C₆₇	—	4	4	no	—
148	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
149	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
150	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
151	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
152	M₁₆×C₆₇	—	2	4	no	—
153	M₁₆×C₆₇	—	2	4	no	—
154	M₁₆×C₆₇	—	4	4	no	—
155	C₂×C₈×C₆₇	—	2	4	no	—
156	C₂×C₈×C₆₇	—	4	4	no	—
157	C₄⋊C₄×C₆₇	—	2	2	no	—
158	C₄⋊C₄×C₆₇	—	2	2	no	—
159	C₄⋊C₄×C₆₇	—	2	1	no	—
160	C₄⋊C₄×C₆₇	—	2	4	no	—
161	C₂²⋊C₄×C₆₇	—	2	2	no	—
162	C₂²⋊C₄×C₆₇	—	2	4	no	—
163	C₂²⋊C₄×C₆₇	—	2	2	no	—
164	C₂²⋊C₄×C₆₇	—	2	1	no	—
165	C₂²⋊C₄×C₆₇	—	2	4	no	—
166	C₂²⋊C₄×C₆₇	—	2	2	no	—
167	C₂²⋊C₄×C₆₇	—	2	4	no	—
168	C₂²⋊C₄×C₆₇	—	4	4	no	—
169	C₄²×C₆₇	—	2	2	no	—
170	C₄²×C₆₇	—	2	1	no	—
171	C₄²×C₆₇	—	2	4	no	—
172	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
173	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
174	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
175	C₂³×C₆₇	—	2	4	no	—
176	C₂³×C₆₇	—	2	2	no	—
177	C₂³×C₆₇	—	2	1	no	—
178	C₂³×C₆₇	—	4	4	no	—
179	C₂³×C₆₇	—	4	4	no	—
180	C₂³×C₆₇	—	4	2	no	—
181	C₂³×C₆₇	—	8	2	no	—
182	C₂³×C₆₇	—	8	4	no	—
183	C₂³×C₆₇	—	8	2	no	—
184	C₂³×C₆₇	—	8	4	no	—
185	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
186	C₂³×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
187	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
188	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
189	Q₈×C₆₇	—	2	2	no	—
190	Q₈×C₆₇	—	2	2	no	—
191	Q₈×C₆₇	—	2	2	no	—
192	D₄×C₆₇	—	2	4	no	—
193	D₄×C₆₇	—	2	4	no	—
194	D₄×C₆₇	—	2	4	no	—
195	D₄×C₆₇	—	2	2	no	—
196	D₄×C₆₇	—	2	4	no	—
197	D₄×C₆₇	—	2	2	no	—
198	D₄×C₆₇	—	2	4	no	—
199	D₄×C₆₇	—	2	2	no	—
200	D₄×C₆₇	—	4	4	no	—
201	D₄×C₆₇	—	4	4	no	—
202	D₄×C₆₇	—	4	4	no	—
203	D₄×C₆₇	—	4	8	no	—
204	D₄×C₆₇	—	4	4	no	—
205	D₄×C₆₇	—	4	4	no	—
206	D₄×C₆₇	—	4	2	no	—
207	D₄×C₆₇	—	4	2	no	—
208	D₄×C₆₇	—	8	4	no	—
209	D₄×C₆₇	—	8	4	no	—
210	D₄×C₆₇	—	8	4	no	—
211	C₂×C₄×C₆₇	—	2	1	no	—
212	C₂×C₄×C₆₇	—	2	4	no	—
213	C₂×C₄×C₆₇	—	2	4	no	—
214	C₂×C₄×C₆₇	—	2	2	no	—
215	C₂×C₄×C₆₇	—	2	1	no	—
216	C₂×C₄×C₆₇	—	2	1	no	—
217	C₂×C₄×C₆₇	—	4	4	no	—
218	C₂×C₄×C₆₇	—	4	4	no	—
219	C₂×C₄×C₆₇	—	4	2	no	—
220	C₂×C₄×C₆₇	—	4	4	no	—
221	C₂×C₄×C₆₇	—	4	2	no	—
222	C₂×C₄×C₆₇	—	4	4	no	—
223	C₂×C₄×C₆₇	—	16	4	no	—
224	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
225	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
226	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
227	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
228	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
229	C₈×C₆₇	—	4	4	no	—
230	C₂²×C₆₇	—	2	2	no	—
231	C₂²×C₆₇	—	2	2	no	—
232	C₂²×C₆₇	—	2	4	no	—
233	C₂²×C₆₇	—	4	1	no	—
234	C₂²×C₆₇	—	4	2	no	—
235	C₂²×C₆₇	—	4	4	no	—
236	C₂²×C₆₇	—	4	2	no	—
237	C₂²×C₆₇	—	4	4	no	—
238	C₂²×C₆₇	—	4	2	no	—
239	C₂²×C₆₇	—	4	2	no	—
240	C₂²×C₆₇	—	4	1	no	—
241	C₂²×C₆₇	—	8	4	no	—
242	C₂²×C₆₇	—	16	2	no	—
243	C₂²×C₆₇	—	16	4	no	—
244	C₂²×C₆₇	—	16	4	no	—
245	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
246	C₂²×C₆₇	#534	1	2	yes	—
247	C₄×C₆₇	—	2	2	no	—
248	C₄×C₆₇	—	2	2	no	—
249	C₄×C₆₇	—	2	2	no	—
250	C₄×C₆₇	—	4	2	no	—
251	C₄×C₆₇	—	4	1	no	—
252	C₄×C₆₇	—	4	1	no	—
253	C₄×C₆₇	—	8	4	no	—
254	C₄×C₆₇	#518	1	2	yes	—
255	C₄×C₆₇	#534	1	2	yes	—
256	#21782	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
257	C₂×C₆₇	—	2	1	no	—
258	C₂×C₆₇	—	2	2	no	—
259	C₂×C₆₇	—	4	4	no	—
260	C₂×C₆₇	—	4	2	no	—
261	C₂×C₆₇	—	4	2	no	—
262	C₂×C₆₇	—	8	2	no	—
263	C₂×C₆₇	—	8	1	no	—
264	C₂×C₆₇	#2308	1	1	yes	characteristic
265	#3498	C₂×C₆₇	1	1	yes	characteristic
266	#2888	C₂×C₆₇	1	1	yes	characteristic, complemented
267	#2883	C₂×C₆₇	1	1	yes	characteristic, complemented
268	#1911	C₂×C₆₇	1	2	yes	complemented
269	#1910	C₂×C₆₇	1	2	yes	complemented
270	#1887	C₂×C₆₇	1	2	yes	complemented
271	#1884	C₂×C₆₇	1	2	yes	complemented
272	#1827	C₂×C₆₇	1	2	yes	complemented
273	#1795	C₂×C₆₇	1	2	yes	complemented
274	C₆₇	#21782	1	1	yes	characteristic, direct product factor, Sylow
275	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	1	yes	characteristic
276	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	1	yes	characteristic
277	#568	C₂²×C₆₇	1	2	yes	complemented
278	#550	—	2	2	no	—
279	D₄²	—	2	4	no	—
280	D₄²	—	2	2	no	—
281	#535	C₄×C₆₇	1	2	yes	complemented
282	#535	C₄×C₆₇	1	2	yes	complemented
283	#535	C₂²×C₆₇	1	1	yes	characteristic
284	#535	C₂²×C₆₇	1	1	yes	characteristic
285	#532	C₄×C₆₇	1	2	yes	complemented
286	#532	C₂²×C₆₇	1	1	yes	characteristic
287	C₂×#127	C₄×C₆₇	1	2	yes	complemented
288	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
289	#518	C₂²×C₆₇	1	2	yes	complemented
290	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
291	#421	C₂²×C₆₇	1	4	yes	complemented
292	#418	C₂²×C₆₇	1	4	yes	complemented
293	#413	—	2	2	no	—
294	#413	C₂²×C₆₇	1	2	yes	—
295	C₂×#100	—	2	2	no	—
296	C₂×#100	C₂²×C₆₇	1	2	yes	complemented
297	#410	C₂²×C₆₇	1	2	yes	—
298	#410	C₂²×C₆₇	1	2	yes	complemented
299	C₂×#99	—	2	4	no	—
300	#339	C₂²×C₆₇	1	4	yes	complemented
301	H(3,ℤ/4)	C₂²×C₆₇	1	4	yes	complemented
302	H(5,𝔽₂)	—	4	4	no	—
303	H(5,𝔽₂)	—	4	4	no	—
304	C₂×#41	—	2	1	no	—
305	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
306	C₂×#41	C₂×C₄×C₆₇	1	2	yes	complemented
307	C₂×#41	D₄×C₆₇	1	2	yes	—
308	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
309	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
310	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
311	C₂²×D₄	—	2	4	no	—
312	C₂²×D₄	—	2	2	no	—
313	C₂²×D₄	—	2	2	no	—
314	C₂²×D₄	—	2	1	no	—
315	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
316	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	complemented
317	C₂²×D₄	D₄×C₆₇	1	4	yes	—
318	C₂²×D₄	D₄×C₆₇	1	2	yes	—
319	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
320	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
321	#131	—	4	4	no	—
322	C₂×M₁₆	—	2	4	no	—
323	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
324	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic, complemented
325	#127	—	2	4	no	—
326	#127	—	2	2	no	—
327	#127	C₂×C₄×C₆₇	1	2	yes	—
328	#127	C₂×C₄×C₆₇	1	2	yes	—
329	#127	C₂×C₄×C₆₇	1	2	yes	—
330	#127	C₂×C₄×C₆₇	1	1	yes	characteristic, complemented
331	#127	C₂×C₄×C₆₇	1	2	yes	complemented
332	#124	C₂×C₄×C₆₇	1	2	yes	—
333	#124	C₂×C₄×C₆₇	1	2	yes	complemented
334	#124	C₂×C₄×C₆₇	1	1	yes	characteristic, complemented
335	#124	C₂×C₄×C₆₇	1	1	yes	characteristic, complemented
336	#121	—	2	4	no	—
337	#121	—	2	4	no	—
338	#121	—	2	2	no	—
339	#121	—	2	2	no	—
340	#121	—	2	4	no	—
341	#121	—	2	2	no	—
342	#120	—	2	4	no	—
343	#120	—	2	2	no	—
344	#120	—	2	4	no	—
345	#120	—	2	2	no	—
346	C₄×Q₈	—	2	1	no	—
347	C₄×D₄	—	2	2	no	—
348	C₄×D₄	—	2	1	no	—
349	C₄×D₄	—	2	4	no	—
350	C₄×D₄	—	2	2	no	—
351	C₄×D₄	—	4	4	no	—
352	#117	C₂×C₄×C₆₇	1	2	yes	complemented
353	#117	C₂³×C₆₇	1	4	yes	—
354	#117	C₂³×C₆₇	1	1	yes	characteristic
355	C₂×C₂²⋊C₄	—	2	4	no	—
356	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	complemented
357	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
358	#104	—	4	4	no	—
359	#102	—	4	4	no	—
360	#101	—	2	2	no	—
361	#100	—	2	2	no	—
362	#100	—	2	4	no	—
363	#99	—	2	4	no	—
364	#99	—	2	4	no	—
365	C₂⁴	—	2	2	no	—
366	C₂⁴	—	2	1	no	—
367	C₂⁴	C₂²⋊C₄×C₆₇	1	4	yes	—
368	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
369	#41	—	2	4	no	—
370	#41	—	2	4	no	—
371	#41	—	2	4	no	—
372	#41	—	4	4	no	—
373	#41	—	4	4	no	—
374	#41	—	4	4	no	—
375	#41	—	4	4	no	—
376	C₂×Q₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
377	C₂×Q₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
378	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
379	C₂×D₄	—	2	4	no	—
380	C₂×D₄	—	2	2	no	—
381	C₂×D₄	—	2	2	no	—
382	C₂×D₄	—	2	1	no	—
383	C₂×D₄	—	2	4	no	—
384	C₂×D₄	—	2	4	no	—
385	C₂×D₄	—	2	4	no	—
386	C₂×D₄	—	2	2	no	—
387	C₂×D₄	—	2	1	no	—
388	C₂×D₄	—	2	4	no	—
389	C₂×D₄	—	2	4	no	—
390	C₂×D₄	—	4	4	no	—
391	C₂×D₄	—	4	4	no	—
392	C₂×D₄	—	4	4	no	—
393	C₂×D₄	—	4	8	no	—
394	C₂×D₄	—	4	8	no	—
395	C₂×D₄	—	4	4	no	—
396	C₂×D₄	—	8	4	no	—
397	C₂×D₄	—	8	8	no	—
398	C₂×D₄	C₂²⋊C₄×C₆₇	1	4	yes	—
399	C₂×D₄	C₂²⋊C₄×C₆₇	1	4	yes	—
400	C₂×D₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
401	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
402	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
403	C₂×D₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
404	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
405	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
406	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
407	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
408	C₂²×C₄	—	2	2	no	—
409	C₂²×C₄	—	2	1	no	—
410	C₂²×C₄	—	2	2	no	—
411	C₂²×C₄	—	2	1	no	—
412	C₂²×C₄	—	4	4	no	—
413	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
414	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
415	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
416	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
417	M₁₆	—	2	4	no	—
418	M₁₆	—	2	4	no	—
419	M₁₆	—	4	4	no	—
420	C₂×C₈	—	2	4	no	—
421	C₂×C₈	—	4	4	no	—
422	C₄⋊C₄	—	2	2	no	—
423	C₄⋊C₄	—	2	2	no	—
424	C₄⋊C₄	—	2	1	no	—
425	C₄⋊C₄	—	2	4	no	—
426	C₂²⋊C₄	—	2	2	no	—
427	C₂²⋊C₄	—	2	4	no	—
428	C₂²⋊C₄	—	2	2	no	—
429	C₂²⋊C₄	—	2	1	no	—
430	C₂²⋊C₄	—	2	4	no	—
431	C₂²⋊C₄	—	2	2	no	—
432	C₂²⋊C₄	—	2	4	no	—
433	C₂²⋊C₄	—	4	4	no	—
434	C₄²	—	2	2	no	—
435	C₄²	—	2	1	no	—
436	C₄²	—	2	4	no	—
437	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
438	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
439	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
440	C₂³	—	2	4	no	—
441	C₂³	—	2	2	no	—
442	C₂³	—	2	1	no	—
443	C₂³	—	4	4	no	—
444	C₂³	—	4	4	no	—
445	C₂³	—	4	2	no	—
446	C₂³	—	8	2	no	—
447	C₂³	—	8	4	no	—
448	C₂³	—	8	2	no	—
449	C₂³	—	8	4	no	—
450	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
451	C₂³	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
452	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
453	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
454	Q₈	—	2	2	no	—
455	Q₈	—	2	2	no	—
456	Q₈	—	2	2	no	—
457	D₄	—	2	4	no	—
458	D₄	—	2	4	no	—
459	D₄	—	2	4	no	—
460	D₄	—	2	2	no	—
461	D₄	—	2	4	no	—
462	D₄	—	2	2	no	—
463	D₄	—	2	4	no	—
464	D₄	—	2	2	no	—
465	D₄	—	4	4	no	—
466	D₄	—	4	4	no	—
467	D₄	—	4	4	no	—
468	D₄	—	4	8	no	—
469	D₄	—	4	4	no	—
470	D₄	—	4	4	no	—
471	D₄	—	4	2	no	—
472	D₄	—	4	2	no	—
473	D₄	—	8	4	no	—
474	D₄	—	8	4	no	—
475	D₄	—	8	4	no	—
476	C₂×C₄	—	2	1	no	—
477	C₂×C₄	—	2	4	no	—
478	C₂×C₄	—	2	4	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	1	no	—
481	C₂×C₄	—	2	1	no	—
482	C₂×C₄	—	4	4	no	—
483	C₂×C₄	—	4	4	no	—
484	C₂×C₄	—	4	2	no	—
485	C₂×C₄	—	4	4	no	—
486	C₂×C₄	—	4	2	no	—
487	C₂×C₄	—	4	4	no	—
488	C₂×C₄	—	16	4	no	—
489	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
490	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
491	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
492	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
493	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
494	C₈	—	4	4	no	—
495	C₂²	—	2	2	no	—
496	C₂²	—	2	2	no	—
497	C₂²	—	2	4	no	—
498	C₂²	—	4	1	no	—
499	C₂²	—	4	2	no	—
500	C₂²	—	4	4	no	—
501	C₂²	—	4	2	no	—
502	C₂²	—	4	4	no	—
503	C₂²	—	4	2	no	—
504	C₂²	—	4	2	no	—
505	C₂²	—	4	1	no	—
506	C₂²	—	8	4	no	—
507	C₂²	—	16	2	no	—
508	C₂²	—	16	4	no	—
509	C₂²	—	16	4	no	—
510	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
511	C₂²	#534×C₆₇	1	2	yes	—
512	C₄	—	2	2	no	—
513	C₄	—	2	2	no	—
514	C₄	—	2	2	no	—
515	C₄	—	4	2	no	—
516	C₄	—	4	1	no	—
517	C₄	—	4	1	no	—
518	C₄	—	8	4	no	—
519	C₄	#518×C₆₇	1	2	yes	—
520	C₄	#534×C₆₇	1	2	yes	—
521	C₂	—	2	1	no	—
522	C₂	—	2	2	no	—
523	C₂	—	4	4	no	—
524	C₂	—	4	2	no	—
525	C₂	—	4	2	no	—
526	C₂	—	8	2	no	—
527	C₂	—	8	1	no	—
528	C₂	C₆₇×#2308	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₇	#2308
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₆₇×#3498	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2888	C₂	semidirect
3. C₆₇×#2883	C₂	semidirect
4. C₆₇×#1911	C₂	semidirect
5. C₆₇×#1910	C₂	semidirect
6. C₆₇×#1887	C₂	semidirect
7. C₆₇×#1884	C₂	semidirect
8. C₆₇×#1827	C₂	semidirect
9. C₆₇×#1795	C₂	semidirect
12. #568×C₆₇	C₂²	semidirect
16. #535×C₆₇	C₄	semidirect
17. #535×C₆₇	C₄	semidirect
20. #532×C₆₇	C₄	semidirect
22. C₂×#127×C₆₇	C₄	semidirect
24. #518×C₆₇	C₂²	semidirect
26. #421×C₆₇	C₂²	semidirect
27. #418×C₆₇	C₂²	semidirect
31. C₂×#100×C₆₇	C₂²	semidirect
33. #410×C₆₇	C₂²	semidirect
35. #339×C₆₇	C₂²	semidirect
36. H(3,ℤ/4)×C₆₇	C₂²	semidirect
41. C₂×#41×C₆₇	C₂×C₄	semidirect
51. C₂²×D₄×C₆₇	C₂×C₄	semidirect
59. C₄⋊Q₈×C₆₇	C₂×C₄	semidirect
65. #127×C₆₇	C₂×C₄	semidirect
66. #127×C₆₇	C₂×C₄	semidirect
68. #124×C₆₇	C₂×C₄	semidirect
69. #124×C₆₇	C₂×C₄	semidirect
70. #124×C₆₇	C₂×C₄	semidirect
87. #117×C₆₇	C₂×C₄	semidirect
91. C₂×C₄²×C₆₇	C₂×C₄	semidirect
256. #21782	C₆₇	direct (N × H)
266. #2888	C₂×C₆₇	semidirect
267. #2883	C₂×C₆₇	semidirect
268. #1911	C₂×C₆₇	semidirect
269. #1910	C₂×C₆₇	semidirect
270. #1887	C₂×C₆₇	semidirect
271. #1884	C₂×C₆₇	semidirect
272. #1827	C₂×C₆₇	semidirect
273. #1795	C₂×C₆₇	semidirect
274. C₆₇	#21782	direct (N × H)
277. #568	C₂²×C₆₇	semidirect
281. #535	C₄×C₆₇	semidirect
282. #535	C₄×C₆₇	semidirect
285. #532	C₄×C₆₇	semidirect
287. C₂×#127	C₄×C₆₇	semidirect
289. #518	C₂²×C₆₇	semidirect
291. #421	C₂²×C₆₇	semidirect
292. #418	C₂²×C₆₇	semidirect
296. C₂×#100	C₂²×C₆₇	semidirect
298. #410	C₂²×C₆₇	semidirect
300. #339	C₂²×C₆₇	semidirect
301. H(3,ℤ/4)	C₂²×C₆₇	semidirect
306. C₂×#41	C₂×C₄×C₆₇	semidirect
316. C₂²×D₄	C₂×C₄×C₆₇	semidirect
324. C₄⋊Q₈	C₂×C₄×C₆₇	semidirect
330. #127	C₂×C₄×C₆₇	semidirect
331. #127	C₂×C₄×C₆₇	semidirect
333. #124	C₂×C₄×C₆₇	semidirect
334. #124	C₂×C₄×C₆₇	semidirect
335. #124	C₂×C₄×C₆₇	semidirect
352. #117	C₂×C₄×C₆₇	semidirect
356. C₂×C₄²	C₂×C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3498	2
C₆₇×#2888	2
C₆₇×#2883	2
C₆₇×#1911	2
C₆₇×#1910	2
C₆₇×#1887	2
C₆₇×#1884	2
C₆₇×#1827	2
C₆₇×#1795	2
#21782	67

Subgroups — C67×#21782 #30701033

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21782 #30701033