Subgroups — C₆₇×#32925 #30712176

Order 17152.

1352 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×2₋¹⁺⁴×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3040	C₂	1	2	yes	maximal, complemented
3	C₂×#456×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#454×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#2545	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1888	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1886	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×#1801	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×2₋¹⁺⁴×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×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
18	#520×C₆₇	—	2	2	no	—
19	#520×C₆₇	—	2	2	no	—
20	#520×C₆₇	C₂²	1	2	yes	—
21	#519×C₆₇	—	2	2	no	—
22	#519×C₆₇	C₂²	1	1	yes	characteristic
23	C₄×#41×C₆₇	—	2	2	no	—
24	C₄×#41×C₆₇	—	2	2	no	—
25	C₄×#41×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₆₇	—	2	2	no	—
27	C₂×C₄×Q₈×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	#456×C₆₇	—	2	4	no	—
33	#454×C₆₇	—	2	4	no	—
34	#423×C₆₇	C₂²	1	2	yes	—
35	C₂×#104×C₆₇	C₂²	1	2	yes	—
36	C₂×#104×C₆₇	C₂²	1	2	yes	—
37	#419×C₆₇	C₂²	1	2	yes	—
38	#418×C₆₇	C₂²	1	2	yes	—
39	C₂×#101×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×#101×C₆₇	C₂²	1	1	yes	characteristic
41	#396×C₆₇	—	2	2	no	—
42	#391×C₆₇	C₂²	1	1	yes	characteristic
43	#390×C₆₇	C₂²	1	1	yes	characteristic
44	#388×C₆₇	—	2	2	no	—
45	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
46	#343×C₆₇	C₂²	1	2	yes	—
47	#339×C₆₇	C₂²	1	2	yes	—
48	#339×C₆₇	C₂²	1	2	yes	—
49	2₋¹⁺⁴×C₆₇	—	2	4	no	—
50	2₋¹⁺⁴×C₆₇	—	2	2	no	—
51	2₋¹⁺⁴×C₆₇	D₄	1	4	yes	—
52	C₂×#41×C₆₇	—	2	2	no	—
53	C₂×#41×C₆₇	—	2	2	no	—
54	C₂×#41×C₆₇	C₂³	1	2	yes	—
55	C₂²×Q₈×C₆₇	—	2	2	no	—
56	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
57	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
58	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
59	#137×C₆₇	—	2	4	no	—
60	#137×C₆₇	—	2	4	no	—
61	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
62	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
63	C₂×Q₁₆×C₆₇	—	2	2	no	—
64	C₂×SD₁₆×C₆₇	—	2	2	no	—
65	C₂×SD₁₆×C₆₇	—	2	2	no	—
66	C₂×D₈×C₆₇	—	2	2	no	—
67	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
68	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₆₇	—	2	2	no	—
70	C₄⋊Q₈×C₆₇	—	2	2	no	—
71	#124×C₆₇	—	2	2	no	—
72	#124×C₆₇	—	2	2	no	—
73	#124×C₆₇	—	2	2	no	—
74	#124×C₆₇	—	2	2	no	—
75	C₄×Q₈×C₆₇	—	2	2	no	—
76	C₄×Q₈×C₆₇	—	2	4	no	—
77	C₄×Q₈×C₆₇	—	2	2	no	—
78	C₄×Q₈×C₆₇	—	2	2	no	—
79	C₄×Q₈×C₆₇	—	2	2	no	—
80	C₄×Q₈×C₆₇	—	2	2	no	—
81	C₄×Q₈×C₆₇	—	2	2	no	—
82	C₄×Q₈×C₆₇	—	2	2	no	—
83	C₄×Q₈×C₆₇	D₄	1	2	yes	—
84	C₄×Q₈×C₆₇	D₄	1	2	yes	—
85	C₄×D₄×C₆₇	—	2	2	no	—
86	C₄×D₄×C₆₇	—	2	4	no	—
87	C₄×D₄×C₆₇	—	2	2	no	—
88	C₄×D₄×C₆₇	—	2	2	no	—
89	C₄×D₄×C₆₇	—	2	2	no	—
90	C₄×D₄×C₆₇	—	2	4	no	—
91	C₄×D₄×C₆₇	—	2	2	no	—
92	C₄×D₄×C₆₇	D₄	1	2	yes	—
93	C₄×D₄×C₆₇	D₄	1	2	yes	—
94	#117×C₆₇	—	2	4	no	—
95	#117×C₆₇	—	2	4	no	—
96	#117×C₆₇	—	2	2	no	—
97	#117×C₆₇	—	2	2	no	—
98	#117×C₆₇	—	2	2	no	—
99	#117×C₆₇	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
101	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
102	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
103	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
104	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
105	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
106	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
107	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
108	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
109	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
110	C₂×C₄²×C₆₇	—	2	4	no	—
111	C₂×C₄²×C₆₇	—	2	2	no	—
112	C₂×C₄²×C₆₇	D₄	1	2	yes	—
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	#101×C₆₇	—	2	2	no	—
125	#101×C₆₇	—	2	2	no	—
126	#95×C₆₇	—	2	1	no	—
127	#95×C₆₇	—	2	2	no	—
128	C₂⁴×C₆₇	—	2	1	no	—
129	#41×C₆₇	—	2	8	no	—
130	#41×C₆₇	—	2	8	no	—
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	8	no	—
139	C₂×Q₈×C₆₇	—	2	8	no	—
140	C₂×Q₈×C₆₇	—	2	2	no	—
141	C₂×Q₈×C₆₇	—	2	2	no	—
142	C₂×Q₈×C₆₇	—	2	2	no	—
143	C₂×Q₈×C₆₇	—	2	1	no	—
144	C₂×Q₈×C₆₇	—	2	1	no	—
145	C₂×Q₈×C₆₇	—	2	2	no	—
146	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
147	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
148	C₂×Q₈×C₆₇	#41	1	2	yes	—
149	C₂×D₄×C₆₇	—	2	2	no	—
150	C₂×D₄×C₆₇	—	2	2	no	—
151	C₂×D₄×C₆₇	—	2	2	no	—
152	C₂×D₄×C₆₇	—	2	1	no	—
153	C₂×D₄×C₆₇	—	2	1	no	—
154	C₂×D₄×C₆₇	—	2	2	no	—
155	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
156	C₂²×C₄×C₆₇	—	2	4	no	—
157	C₂²×C₄×C₆₇	—	2	2	no	—
158	C₂²×C₄×C₆₇	—	2	1	no	—
159	C₂²×C₄×C₆₇	—	2	4	no	—
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	1	no	—
164	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
165	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
166	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
169	Q₁₆×C₆₇	—	4	4	no	—
170	SD₁₆×C₆₇	—	4	4	no	—
171	SD₁₆×C₆₇	—	4	4	no	—
172	D₈×C₆₇	—	4	4	no	—
173	M₁₆×C₆₇	—	2	2	no	—
174	M₁₆×C₆₇	—	2	2	no	—
175	M₁₆×C₆₇	—	2	2	no	—
176	M₁₆×C₆₇	—	2	2	no	—
177	C₂×C₈×C₆₇	—	2	2	no	—
178	C₂×C₈×C₆₇	—	2	2	no	—
179	C₄⋊C₄×C₆₇	—	2	4	no	—
180	C₄⋊C₄×C₆₇	—	2	2	no	—
181	C₄⋊C₄×C₆₇	—	2	2	no	—
182	C₄⋊C₄×C₆₇	—	2	2	no	—
183	C₄⋊C₄×C₆₇	—	2	4	no	—
184	C₄⋊C₄×C₆₇	—	2	2	no	—
185	C₄⋊C₄×C₆₇	—	2	2	no	—
186	C₄⋊C₄×C₆₇	—	4	2	no	—
187	C₄⋊C₄×C₆₇	—	4	4	no	—
188	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
189	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
190	C₂²⋊C₄×C₆₇	—	2	4	no	—
191	C₂²⋊C₄×C₆₇	—	2	2	no	—
192	C₂²⋊C₄×C₆₇	—	2	2	no	—
193	C₂²⋊C₄×C₆₇	—	2	2	no	—
194	C₂²⋊C₄×C₆₇	—	4	4	no	—
195	C₂²⋊C₄×C₆₇	—	4	1	no	—
196	C₂²⋊C₄×C₆₇	—	4	1	no	—
197	C₂²⋊C₄×C₆₇	—	4	2	no	—
198	C₂²⋊C₄×C₆₇	—	4	4	no	—
199	C₄²×C₆₇	—	2	4	no	—
200	C₄²×C₆₇	—	2	4	no	—
201	C₄²×C₆₇	—	2	2	no	—
202	C₄²×C₆₇	—	2	2	no	—
203	C₄²×C₆₇	—	2	2	no	—
204	C₄²×C₆₇	—	2	2	no	—
205	C₄²×C₆₇	—	2	2	no	—
206	C₄²×C₆₇	—	2	1	no	—
207	C₄²×C₆₇	—	2	1	no	—
208	C₄²×C₆₇	—	2	1	no	—
209	C₄²×C₆₇	—	2	1	no	—
210	C₄²×C₆₇	C₂×D₄	1	2	yes	—
211	C₄²×C₆₇	C₂×D₄	1	2	yes	—
212	C₂³×C₆₇	—	2	2	no	—
213	C₂³×C₆₇	—	4	2	no	—
214	C₂³×C₆₇	—	4	2	no	—
215	C₂³×C₆₇	—	4	1	no	—
216	C₂³×C₆₇	—	4	1	no	—
217	C₂³×C₆₇	—	4	1	no	—
218	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
219	Q₈×C₆₇	—	2	8	no	—
220	Q₈×C₆₇	—	2	4	no	—
221	Q₈×C₆₇	—	2	2	no	—
222	Q₈×C₆₇	—	2	4	no	—
223	Q₈×C₆₇	—	4	2	no	—
224	Q₈×C₆₇	—	4	2	no	—
225	Q₈×C₆₇	#121	1	4	yes	—
226	D₄×C₆₇	—	2	8	no	—
227	D₄×C₆₇	—	2	4	no	—
228	D₄×C₆₇	—	2	4	no	—
229	D₄×C₆₇	—	2	2	no	—
230	D₄×C₆₇	—	4	2	no	—
231	D₄×C₆₇	—	4	2	no	—
232	D₄×C₆₇	#121	1	4	yes	—
233	C₂×C₄×C₆₇	—	2	16	no	—
234	C₂×C₄×C₆₇	—	2	2	no	—
235	C₂×C₄×C₆₇	—	2	2	no	—
236	C₂×C₄×C₆₇	—	2	2	no	—
237	C₂×C₄×C₆₇	—	2	4	no	—
238	C₂×C₄×C₆₇	—	2	4	no	—
239	C₂×C₄×C₆₇	—	2	2	no	—
240	C₂×C₄×C₆₇	—	2	2	no	—
241	C₂×C₄×C₆₇	—	2	2	no	—
242	C₂×C₄×C₆₇	—	2	2	no	—
243	C₂×C₄×C₆₇	—	4	1	no	—
244	C₂×C₄×C₆₇	—	4	1	no	—
245	C₂×C₄×C₆₇	—	4	4	no	—
246	C₂×C₄×C₆₇	—	4	1	no	—
247	C₂×C₄×C₆₇	—	4	1	no	—
248	C₂×C₄×C₆₇	—	4	2	no	—
249	C₂×C₄×C₆₇	—	4	1	no	—
250	C₂×C₄×C₆₇	—	4	1	no	—
251	C₂×C₄×C₆₇	—	4	1	no	—
252	C₂×C₄×C₆₇	—	4	1	no	—
253	C₂×C₄×C₆₇	—	4	1	no	—
254	C₂×C₄×C₆₇	—	4	1	no	—
255	C₂×C₄×C₆₇	—	8	2	no	—
256	C₂×C₄×C₆₇	—	8	4	no	—
257	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
258	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
259	C₂×C₄×C₆₇	#120	1	2	yes	—
260	C₂×C₄×C₆₇	#121	1	4	yes	—
261	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
262	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
263	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
264	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
265	C₈×C₆₇	—	4	2	no	—
266	C₈×C₆₇	—	4	2	no	—
267	C₂²×C₆₇	—	2	1	no	—
268	C₂²×C₆₇	—	2	1	no	—
269	C₂²×C₆₇	—	2	4	no	—
270	C₂²×C₆₇	—	4	2	no	—
271	C₂²×C₆₇	—	4	2	no	—
272	C₂²×C₆₇	—	8	2	no	—
273	C₂²×C₆₇	—	8	2	no	—
274	C₂²×C₆₇	—	8	1	no	—
275	C₂²×C₆₇	—	8	1	no	—
276	C₂²×C₆₇	—	8	1	no	—
277	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
278	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
279	C₂²×C₆₇	#525	1	1	yes	characteristic
280	C₄×C₆₇	—	2	4	no	—
281	C₄×C₆₇	—	2	4	no	—
282	C₄×C₆₇	—	2	2	no	—
283	C₄×C₆₇	—	2	1	no	—
284	C₄×C₆₇	—	2	1	no	—
285	C₄×C₆₇	—	4	2	no	—
286	C₄×C₆₇	—	4	2	no	—
287	C₄×C₆₇	—	4	2	no	—
288	C₄×C₆₇	—	4	2	no	—
289	C₄×C₆₇	—	8	1	no	—
290	C₄×C₆₇	—	8	1	no	—
291	C₄×C₆₇	C₂×#121	1	2	yes	—
292	C₄×C₆₇	#525	1	2	yes	—
293	#32925	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
294	C₂×C₆₇	—	2	1	no	—
295	C₂×C₆₇	—	2	1	no	—
296	C₂×C₆₇	—	4	4	no	—
297	C₂×C₆₇	—	16	1	no	—
298	C₂×C₆₇	#2847	1	1	yes	characteristic
299	C₂×C₆₇	#3018	1	1	yes	characteristic
300	C₂×C₆₇	#3020	1	1	yes	characteristic
301	C₄×2₋¹⁺⁴	C₂×C₆₇	1	1	yes	characteristic, complemented
302	#3040	C₂×C₆₇	1	2	yes	complemented
303	C₂×#456	C₂×C₆₇	1	1	yes	characteristic, complemented
304	C₂×#454	C₂×C₆₇	1	1	yes	characteristic
305	#2545	C₂×C₆₇	1	1	yes	characteristic, complemented
306	#1888	C₂×C₆₇	1	1	yes	characteristic, complemented
307	#1886	C₂×C₆₇	1	1	yes	characteristic, complemented
308	#1883	C₂×C₆₇	1	2	yes	complemented
309	#1882	C₂×C₆₇	1	2	yes	complemented
310	#1806	C₂×C₆₇	1	2	yes	complemented
311	#1801	C₂×C₆₇	1	1	yes	characteristic, complemented
312	C₆₇	#32925	1	1	yes	characteristic, direct product factor, Sylow
313	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	1	yes	characteristic
314	C₂×#137	C₂²×C₆₇	1	2	yes	—
315	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	2	yes	—
316	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
317	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
318	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
319	#520	—	2	2	no	—
320	#520	—	2	2	no	—
321	#520	C₂²×C₆₇	1	2	yes	—
322	#519	—	2	2	no	—
323	#519	C₂²×C₆₇	1	1	yes	characteristic
324	C₄×#41	—	2	2	no	—
325	C₄×#41	—	2	2	no	—
326	C₄×#41	C₂²×C₆₇	1	2	yes	—
327	C₂×C₄×Q₈	—	2	2	no	—
328	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
329	#464	—	2	2	no	—
330	#464	—	2	2	no	—
331	#463	—	2	2	no	—
332	#463	—	2	2	no	—
333	#456	—	2	4	no	—
334	#454	—	2	4	no	—
335	#423	C₂²×C₆₇	1	2	yes	—
336	C₂×#104	C₂²×C₆₇	1	2	yes	—
337	C₂×#104	C₂²×C₆₇	1	2	yes	—
338	#419	C₂²×C₆₇	1	2	yes	—
339	#418	C₂²×C₆₇	1	2	yes	—
340	C₂×#101	C₂²×C₆₇	1	1	yes	characteristic
341	C₂×#101	C₂²×C₆₇	1	1	yes	characteristic
342	#396	—	2	2	no	—
343	#391	C₂²×C₆₇	1	1	yes	characteristic
344	#390	C₂²×C₆₇	1	1	yes	characteristic
345	#388	—	2	2	no	—
346	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
347	#343	C₂²×C₆₇	1	2	yes	—
348	#339	C₂²×C₆₇	1	2	yes	—
349	#339	C₂²×C₆₇	1	2	yes	—
350	2₋¹⁺⁴	—	2	4	no	—
351	2₋¹⁺⁴	—	2	2	no	—
352	2₋¹⁺⁴	D₄×C₆₇	1	4	yes	—
353	C₂×#41	—	2	2	no	—
354	C₂×#41	—	2	2	no	—
355	C₂×#41	C₂³×C₆₇	1	2	yes	—
356	C₂²×Q₈	—	2	2	no	—
357	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
358	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
359	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
360	#137	—	2	4	no	—
361	#137	—	2	4	no	—
362	AGL(1,ℤ/8)	—	2	4	no	—
363	AGL(1,ℤ/8)	—	2	4	no	—
364	C₂×Q₁₆	—	2	2	no	—
365	C₂×SD₁₆	—	2	2	no	—
366	C₂×SD₁₆	—	2	2	no	—
367	C₂×D₈	—	2	2	no	—
368	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
369	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
370	C₄⋊Q₈	—	2	2	no	—
371	C₄⋊Q₈	—	2	2	no	—
372	#124	—	2	2	no	—
373	#124	—	2	2	no	—
374	#124	—	2	2	no	—
375	#124	—	2	2	no	—
376	C₄×Q₈	—	2	2	no	—
377	C₄×Q₈	—	2	4	no	—
378	C₄×Q₈	—	2	2	no	—
379	C₄×Q₈	—	2	2	no	—
380	C₄×Q₈	—	2	2	no	—
381	C₄×Q₈	—	2	2	no	—
382	C₄×Q₈	—	2	2	no	—
383	C₄×Q₈	—	2	2	no	—
384	C₄×Q₈	D₄×C₆₇	1	2	yes	—
385	C₄×Q₈	D₄×C₆₇	1	2	yes	—
386	C₄×D₄	—	2	2	no	—
387	C₄×D₄	—	2	4	no	—
388	C₄×D₄	—	2	2	no	—
389	C₄×D₄	—	2	2	no	—
390	C₄×D₄	—	2	2	no	—
391	C₄×D₄	—	2	4	no	—
392	C₄×D₄	—	2	2	no	—
393	C₄×D₄	D₄×C₆₇	1	2	yes	—
394	C₄×D₄	D₄×C₆₇	1	2	yes	—
395	#117	—	2	4	no	—
396	#117	—	2	4	no	—
397	#117	—	2	2	no	—
398	#117	—	2	2	no	—
399	#117	—	2	2	no	—
400	#117	C₂³×C₆₇	1	2	yes	—
401	C₂×C₄⋊C₄	—	2	4	no	—
402	C₂×C₄⋊C₄	—	2	2	no	—
403	C₂×C₄⋊C₄	—	2	2	no	—
404	C₂×C₄⋊C₄	—	2	1	no	—
405	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
406	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
407	C₂×C₂²⋊C₄	—	2	1	no	—
408	C₂×C₂²⋊C₄	—	2	2	no	—
409	C₂×C₂²⋊C₄	—	2	1	no	—
410	C₂×C₂²⋊C₄	—	2	2	no	—
411	C₂×C₄²	—	2	4	no	—
412	C₂×C₄²	—	2	2	no	—
413	C₂×C₄²	D₄×C₆₇	1	2	yes	—
414	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
415	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
416	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
417	C₄⋊C₈	—	2	2	no	—
418	C₄⋊C₈	—	2	2	no	—
419	#104	—	2	4	no	—
420	#104	—	2	4	no	—
421	#103	—	2	2	no	—
422	#103	—	2	2	no	—
423	#102	—	2	2	no	—
424	#102	—	2	2	no	—
425	#101	—	2	2	no	—
426	#101	—	2	2	no	—
427	#95	—	2	1	no	—
428	#95	—	2	2	no	—
429	C₂⁴	—	2	1	no	—
430	#41	—	2	8	no	—
431	#41	—	2	8	no	—
432	#41	—	2	8	no	—
433	#41	—	2	8	no	—
434	#41	—	2	2	no	—
435	#41	—	2	2	no	—
436	#41	C₂×D₄×C₆₇	1	4	yes	—
437	#41	#41×C₆₇	1	4	yes	—
438	C₂×Q₈	—	2	2	no	—
439	C₂×Q₈	—	2	8	no	—
440	C₂×Q₈	—	2	8	no	—
441	C₂×Q₈	—	2	2	no	—
442	C₂×Q₈	—	2	2	no	—
443	C₂×Q₈	—	2	2	no	—
444	C₂×Q₈	—	2	1	no	—
445	C₂×Q₈	—	2	1	no	—
446	C₂×Q₈	—	2	2	no	—
447	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
448	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
449	C₂×Q₈	#41×C₆₇	1	2	yes	—
450	C₂×D₄	—	2	2	no	—
451	C₂×D₄	—	2	2	no	—
452	C₂×D₄	—	2	2	no	—
453	C₂×D₄	—	2	1	no	—
454	C₂×D₄	—	2	1	no	—
455	C₂×D₄	—	2	2	no	—
456	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
457	C₂²×C₄	—	2	4	no	—
458	C₂²×C₄	—	2	2	no	—
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	2	4	no	—
461	C₂²×C₄	—	2	2	no	—
462	C₂²×C₄	—	2	2	no	—
463	C₂²×C₄	—	2	1	no	—
464	C₂²×C₄	—	2	1	no	—
465	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
466	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
467	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
468	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
469	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
470	Q₁₆	—	4	4	no	—
471	SD₁₆	—	4	4	no	—
472	SD₁₆	—	4	4	no	—
473	D₈	—	4	4	no	—
474	M₁₆	—	2	2	no	—
475	M₁₆	—	2	2	no	—
476	M₁₆	—	2	2	no	—
477	M₁₆	—	2	2	no	—
478	C₂×C₈	—	2	2	no	—
479	C₂×C₈	—	2	2	no	—
480	C₄⋊C₄	—	2	4	no	—
481	C₄⋊C₄	—	2	2	no	—
482	C₄⋊C₄	—	2	2	no	—
483	C₄⋊C₄	—	2	2	no	—
484	C₄⋊C₄	—	2	4	no	—
485	C₄⋊C₄	—	2	2	no	—
486	C₄⋊C₄	—	2	2	no	—
487	C₄⋊C₄	—	4	2	no	—
488	C₄⋊C₄	—	4	4	no	—
489	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
490	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
491	C₂²⋊C₄	—	2	4	no	—
492	C₂²⋊C₄	—	2	2	no	—
493	C₂²⋊C₄	—	2	2	no	—
494	C₂²⋊C₄	—	2	2	no	—
495	C₂²⋊C₄	—	4	4	no	—
496	C₂²⋊C₄	—	4	1	no	—
497	C₂²⋊C₄	—	4	1	no	—
498	C₂²⋊C₄	—	4	2	no	—
499	C₂²⋊C₄	—	4	4	no	—
500	C₄²	—	2	4	no	—
501	C₄²	—	2	4	no	—
502	C₄²	—	2	2	no	—
503	C₄²	—	2	2	no	—
504	C₄²	—	2	2	no	—
505	C₄²	—	2	2	no	—
506	C₄²	—	2	2	no	—
507	C₄²	—	2	1	no	—
508	C₄²	—	2	1	no	—
509	C₄²	—	2	1	no	—
510	C₄²	—	2	1	no	—
511	C₄²	C₂×D₄×C₆₇	1	2	yes	—
512	C₄²	C₂×D₄×C₆₇	1	2	yes	—
513	C₂³	—	2	2	no	—
514	C₂³	—	4	2	no	—
515	C₂³	—	4	2	no	—
516	C₂³	—	4	1	no	—
517	C₂³	—	4	1	no	—
518	C₂³	—	4	1	no	—
519	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
520	Q₈	—	2	8	no	—
521	Q₈	—	2	4	no	—
522	Q₈	—	2	2	no	—
523	Q₈	—	2	4	no	—
524	Q₈	—	4	2	no	—
525	Q₈	—	4	2	no	—
526	Q₈	#121×C₆₇	1	4	yes	—
527	D₄	—	2	8	no	—
528	D₄	—	2	4	no	—
529	D₄	—	2	4	no	—
530	D₄	—	2	2	no	—
531	D₄	—	4	2	no	—
532	D₄	—	4	2	no	—
533	D₄	#121×C₆₇	1	4	yes	—
534	C₂×C₄	—	2	16	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	2	4	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	4	1	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	4	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	2	no	—
550	C₂×C₄	—	4	1	no	—
551	C₂×C₄	—	4	1	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	8	2	no	—
557	C₂×C₄	—	8	4	no	—
558	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
559	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
560	C₂×C₄	#120×C₆₇	1	2	yes	—
561	C₂×C₄	#121×C₆₇	1	4	yes	—
562	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
563	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
564	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
565	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
566	C₈	—	4	2	no	—
567	C₈	—	4	2	no	—
568	C₂²	—	2	1	no	—
569	C₂²	—	2	1	no	—
570	C₂²	—	2	4	no	—
571	C₂²	—	4	2	no	—
572	C₂²	—	4	2	no	—
573	C₂²	—	8	2	no	—
574	C₂²	—	8	2	no	—
575	C₂²	—	8	1	no	—
576	C₂²	—	8	1	no	—
577	C₂²	—	8	1	no	—
578	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
579	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
580	C₂²	#525×C₆₇	1	1	yes	characteristic
581	C₄	—	2	2	no	—
582	C₄	—	2	1	no	—
583	C₄	—	2	1	no	—
584	C₄	—	2	4	no	—
585	C₄	—	2	4	no	—
586	C₄	—	4	2	no	—
587	C₄	—	4	2	no	—
588	C₄	—	4	2	no	—
589	C₄	—	4	2	no	—
590	C₄	—	8	1	no	—
591	C₄	—	8	1	no	—
592	C₄	C₂×#121×C₆₇	1	2	yes	—
593	C₄	#525×C₆₇	1	2	yes	—
594	C₂	—	2	1	no	—
595	C₂	—	2	1	no	—
596	C₂	—	4	4	no	—
597	C₂	—	16	1	no	—
598	C₂	C₆₇×#2847	1	1	yes	characteristic
599	C₂	C₆₇×#3018	1	1	yes	characteristic
600	C₂	C₆₇×#3020	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#454×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. C₄×2₋¹⁺⁴×C₆₇	C₂	semidirect
2. C₆₇×#3040	C₂	semidirect
3. C₂×#456×C₆₇	C₂	semidirect
5. C₆₇×#2545	C₂	semidirect
6. C₆₇×#1888	C₂	semidirect
7. C₆₇×#1886	C₂	semidirect
8. C₆₇×#1883	C₂	semidirect
9. C₆₇×#1882	C₂	semidirect
10. C₆₇×#1806	C₂	semidirect
11. C₆₇×#1801	C₂	semidirect
293. #32925	C₆₇	direct (N × H)
301. C₄×2₋¹⁺⁴	C₂×C₆₇	semidirect
302. #3040	C₂×C₆₇	semidirect
303. C₂×#456	C₂×C₆₇	semidirect
305. #2545	C₂×C₆₇	semidirect
306. #1888	C₂×C₆₇	semidirect
307. #1886	C₂×C₆₇	semidirect
308. #1883	C₂×C₆₇	semidirect
309. #1882	C₂×C₆₇	semidirect
310. #1806	C₂×C₆₇	semidirect
311. #1801	C₂×C₆₇	semidirect
312. C₆₇	#32925	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×2₋¹⁺⁴×C₆₇	2
C₆₇×#3040	2
C₂×#456×C₆₇	2
C₂×#454×C₆₇	2
C₆₇×#2545	2
C₆₇×#1888	2
C₆₇×#1886	2
C₆₇×#1883	2
C₆₇×#1882	2
C₆₇×#1806	2
C₆₇×#1801	2
#32925	67

Subgroups — C67×#32925 #30712176

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32925 #30712176