Subgroups — C₆₇×#33070 #30712321

Order 17152.

834 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2864	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄²×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1949	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1949	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1921	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1920	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1919	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1918	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1864	C₂	1	1	yes	maximal, characteristic
10	C₆₇×#1864	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1862	C₂	1	1	yes	maximal, characteristic
12	C₆₇×#1862	C₂	1	1	yes	maximal, characteristic
13	C₆₇×#1831	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₇×#1804	C₂	1	1	yes	maximal, characteristic
15	C₆₇×#1804	C₂	1	1	yes	maximal, characteristic
16	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄²×C₆₇	—	2	1	no	—
20	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
30	#389×C₆₇	C₂²	1	1	yes	characteristic
31	#389×C₆₇	C₂²	1	1	yes	characteristic
32	#389×C₆₇	C₂²	1	1	yes	characteristic
33	#389×C₆₇	C₂²	1	1	yes	characteristic
34	#387×C₆₇	—	2	1	no	—
35	#387×C₆₇	—	2	1	no	—
36	#387×C₆₇	—	2	1	no	—
37	#387×C₆₇	—	2	1	no	—
38	#382×C₆₇	C₂²	1	1	yes	characteristic
39	#382×C₆₇	C₂²	1	1	yes	characteristic
40	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
41	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
42	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
43	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
44	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
45	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
46	C₄³×C₆₇	C₂²	1	1	yes	characteristic
47	#340×C₆₇	C₂²	1	1	yes	characteristic
48	#340×C₆₇	C₂²	1	1	yes	characteristic
49	#340×C₆₇	C₂²	1	1	yes	characteristic
50	#340×C₆₇	C₂²	1	1	yes	characteristic
51	#340×C₆₇	C₂²	1	1	yes	characteristic
52	#340×C₆₇	C₂²	1	1	yes	characteristic
53	#340×C₆₇	C₂²	1	1	yes	characteristic
54	#340×C₆₇	C₂²	1	1	yes	characteristic
55	#336×C₆₇	C₂²	1	1	yes	characteristic
56	#336×C₆₇	C₂²	1	1	yes	characteristic
57	#336×C₆₇	C₂²	1	1	yes	characteristic
58	#336×C₆₇	C₂²	1	1	yes	characteristic
59	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₆₇	—	2	1	no	—
61	C₂³×C₄×C₆₇	—	2	1	no	—
62	C₂³×C₄×C₆₇	—	2	1	no	—
63	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
67	C₄×D₄×C₆₇	—	2	2	no	—
68	C₄×D₄×C₆₇	—	2	2	no	—
69	C₄×D₄×C₆₇	—	2	2	no	—
70	C₄×D₄×C₆₇	—	2	2	no	—
71	C₄×D₄×C₆₇	—	2	2	no	—
72	C₄×D₄×C₆₇	Q₈	1	4	yes	—
73	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
74	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
75	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
76	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
77	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
84	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
85	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
86	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
87	C₂×C₄²×C₆₇	—	2	1	no	—
88	C₂×C₄²×C₆₇	—	2	1	no	—
89	C₂×C₄²×C₆₇	—	2	1	no	—
90	C₂×C₄²×C₆₇	—	4	4	no	—
91	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
92	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
93	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
94	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
95	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
96	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
97	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
98	#107×C₆₇	—	2	2	no	—
99	#107×C₆₇	—	2	2	no	—
100	#106×C₆₇	—	2	2	no	—
101	#106×C₆₇	—	2	2	no	—
102	C₄⋊C₈×C₆₇	—	2	2	no	—
103	C₄⋊C₈×C₆₇	—	2	2	no	—
104	#102×C₆₇	—	2	2	no	—
105	#102×C₆₇	—	2	2	no	—
106	#102×C₆₇	—	2	2	no	—
107	#102×C₆₇	—	2	2	no	—
108	#95×C₆₇	—	2	1	no	—
109	#95×C₆₇	—	2	1	no	—
110	#95×C₆₇	—	2	1	no	—
111	#95×C₆₇	—	2	1	no	—
112	#95×C₆₇	—	2	1	no	—
113	#95×C₆₇	—	2	1	no	—
114	#95×C₆₇	—	2	1	no	—
115	#95×C₆₇	—	2	1	no	—
116	#95×C₆₇	—	2	1	no	—
117	#95×C₆₇	—	2	1	no	—
118	#95×C₆₇	—	2	1	no	—
119	C₂⁴×C₆₇	—	2	1	no	—
120	C₂×D₄×C₆₇	—	2	1	no	—
121	C₂×D₄×C₆₇	—	2	1	no	—
122	C₂×D₄×C₆₇	—	2	1	no	—
123	C₂×D₄×C₆₇	C₂×Q₈	1	2	yes	—
124	C₂×D₄×C₆₇	#41	1	2	yes	—
125	C₂×D₄×C₆₇	#41	1	2	yes	—
126	C₂²×C₄×C₆₇	—	2	4	no	—
127	C₂²×C₄×C₆₇	—	2	4	no	—
128	C₂²×C₄×C₆₇	—	2	1	no	—
129	C₂²×C₄×C₆₇	—	2	1	no	—
130	C₂²×C₄×C₆₇	—	2	1	no	—
131	C₂²×C₄×C₆₇	—	2	4	no	—
132	C₂²×C₄×C₆₇	—	2	1	no	—
133	C₂²×C₄×C₆₇	—	2	1	no	—
134	C₂²×C₄×C₆₇	—	2	1	no	—
135	C₂²×C₄×C₆₇	—	2	1	no	—
136	C₂²×C₄×C₆₇	—	4	2	no	—
137	C₂²×C₄×C₆₇	—	4	2	no	—
138	C₂²×C₄×C₆₇	—	4	2	no	—
139	C₂²×C₄×C₆₇	—	4	2	no	—
140	C₂²×C₄×C₆₇	—	4	2	no	—
141	C₂²×C₄×C₆₇	—	4	2	no	—
142	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
149	C₂×C₈×C₆₇	—	2	1	no	—
150	C₂×C₈×C₆₇	—	2	1	no	—
151	C₂×C₈×C₆₇	—	2	1	no	—
152	C₂×C₈×C₆₇	—	2	1	no	—
153	C₂×C₈×C₆₇	—	2	1	no	—
154	C₂×C₈×C₆₇	—	2	1	no	—
155	C₂×C₈×C₆₇	—	2	1	no	—
156	C₂×C₈×C₆₇	—	2	1	no	—
157	C₂×C₈×C₆₇	—	2	1	no	—
158	C₂×C₈×C₆₇	—	2	1	no	—
159	C₂×C₈×C₆₇	—	2	1	no	—
160	C₂×C₈×C₆₇	—	2	1	no	—
161	C₄⋊C₄×C₆₇	—	2	1	no	—
162	C₄⋊C₄×C₆₇	—	2	1	no	—
163	C₄⋊C₄×C₆₇	—	2	1	no	—
164	C₄⋊C₄×C₆₇	—	2	1	no	—
165	C₄⋊C₄×C₆₇	—	2	1	no	—
166	C₄⋊C₄×C₆₇	—	2	1	no	—
167	C₄⋊C₄×C₆₇	—	2	1	no	—
168	C₄⋊C₄×C₆₇	—	2	1	no	—
169	C₄⋊C₄×C₆₇	—	2	1	no	—
170	C₄⋊C₄×C₆₇	—	2	1	no	—
171	C₄⋊C₄×C₆₇	—	2	1	no	—
172	C₄⋊C₄×C₆₇	—	4	2	no	—
173	C₄⋊C₄×C₆₇	—	4	2	no	—
174	C₄⋊C₄×C₆₇	—	4	2	no	—
175	C₄⋊C₄×C₆₇	—	4	2	no	—
176	C₄⋊C₄×C₆₇	C₂×Q₈	1	2	yes	—
177	C₄⋊C₄×C₆₇	#41	1	2	yes	—
178	C₄⋊C₄×C₆₇	#41	1	2	yes	—
179	C₂²⋊C₄×C₆₇	—	2	4	no	—
180	C₂²⋊C₄×C₆₇	—	2	4	no	—
181	C₂²⋊C₄×C₆₇	—	2	4	no	—
182	C₄²×C₆₇	—	2	2	no	—
183	C₄²×C₆₇	—	2	2	no	—
184	C₄²×C₆₇	—	2	1	no	—
185	C₄²×C₆₇	—	2	1	no	—
186	C₄²×C₆₇	—	2	1	no	—
187	C₄²×C₆₇	—	2	1	no	—
188	C₄²×C₆₇	—	4	1	no	—
189	C₄²×C₆₇	—	4	1	no	—
190	C₄²×C₆₇	—	4	2	no	—
191	C₄²×C₆₇	—	4	2	no	—
192	C₄²×C₆₇	—	4	2	no	—
193	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
194	C₄²×C₆₇	#41	1	2	yes	—
195	C₂³×C₆₇	—	2	2	no	—
196	C₂³×C₆₇	—	2	2	no	—
197	C₂³×C₆₇	—	2	2	no	—
198	C₂³×C₆₇	—	4	1	no	—
199	C₂³×C₆₇	—	4	1	no	—
200	C₂³×C₆₇	—	4	1	no	—
201	C₂³×C₆₇	—	4	1	no	—
202	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
203	D₄×C₆₇	—	2	2	no	—
204	D₄×C₆₇	—	2	2	no	—
205	D₄×C₆₇	—	2	2	no	—
206	D₄×C₆₇	#125	1	4	yes	—
207	C₂×C₄×C₆₇	—	2	1	no	—
208	C₂×C₄×C₆₇	—	2	1	no	—
209	C₂×C₄×C₆₇	—	2	1	no	—
210	C₂×C₄×C₆₇	—	2	1	no	—
211	C₂×C₄×C₆₇	—	2	1	no	—
212	C₂×C₄×C₆₇	—	2	1	no	—
213	C₂×C₄×C₆₇	—	2	1	no	—
214	C₂×C₄×C₆₇	—	2	1	no	—
215	C₂×C₄×C₆₇	—	2	2	no	—
216	C₂×C₄×C₆₇	—	2	1	no	—
217	C₂×C₄×C₆₇	—	2	1	no	—
218	C₂×C₄×C₆₇	—	2	2	no	—
219	C₂×C₄×C₆₇	—	2	2	no	—
220	C₂×C₄×C₆₇	—	2	1	no	—
221	C₂×C₄×C₆₇	—	2	1	no	—
222	C₂×C₄×C₆₇	—	2	1	no	—
223	C₂×C₄×C₆₇	—	2	1	no	—
224	C₂×C₄×C₆₇	—	4	8	no	—
225	C₂×C₄×C₆₇	—	4	8	no	—
226	C₂×C₄×C₆₇	—	4	8	no	—
227	C₂×C₄×C₆₇	—	4	1	no	—
228	C₂×C₄×C₆₇	—	4	1	no	—
229	C₂×C₄×C₆₇	—	4	1	no	—
230	C₂×C₄×C₆₇	—	4	1	no	—
231	C₂×C₄×C₆₇	—	4	1	no	—
232	C₂×C₄×C₆₇	—	4	1	no	—
233	C₂×C₄×C₆₇	—	4	1	no	—
234	C₂×C₄×C₆₇	—	4	1	no	—
235	C₂×C₄×C₆₇	—	4	1	no	—
236	C₂×C₄×C₆₇	—	4	1	no	—
237	C₂×C₄×C₆₇	—	4	1	no	—
238	C₂×C₄×C₆₇	—	4	1	no	—
239	C₂×C₄×C₆₇	—	4	1	no	—
240	C₂×C₄×C₆₇	—	4	1	no	—
241	C₂×C₄×C₆₇	—	4	1	no	—
242	C₂×C₄×C₆₇	—	4	1	no	—
243	C₂×C₄×C₆₇	—	4	1	no	—
244	C₂×C₄×C₆₇	—	4	1	no	—
245	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
246	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
247	C₂×C₄×C₆₇	#135	1	2	yes	—
248	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
249	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
250	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
251	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
252	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
253	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
254	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
255	C₈×C₆₇	—	4	1	no	—
256	C₈×C₆₇	—	4	1	no	—
257	C₈×C₆₇	—	4	1	no	—
258	C₈×C₆₇	—	4	1	no	—
259	C₂²×C₆₇	—	2	4	no	—
260	C₂²×C₆₇	—	4	2	no	—
261	C₂²×C₆₇	—	4	2	no	—
262	C₂²×C₆₇	—	4	2	no	—
263	C₂²×C₆₇	—	4	2	no	—
264	C₂²×C₆₇	—	4	2	no	—
265	C₂²×C₆₇	—	4	2	no	—
266	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
267	C₂²×C₆₇	#525	1	1	yes	characteristic
268	C₂²×C₆₇	#525	1	1	yes	characteristic
269	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
270	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
271	C₂²×C₆₇	#575	1	1	yes	characteristic
272	C₂²×C₆₇	#575	1	1	yes	characteristic
273	C₄×C₆₇	—	2	2	no	—
274	C₄×C₆₇	—	2	2	no	—
275	C₄×C₆₇	—	2	2	no	—
276	C₄×C₆₇	—	2	2	no	—
277	C₄×C₆₇	—	4	2	no	—
278	C₄×C₆₇	—	4	1	no	—
279	C₄×C₆₇	—	4	2	no	—
280	C₄×C₆₇	—	4	1	no	—
281	C₄×C₆₇	—	4	2	no	—
282	C₄×C₆₇	—	8	1	no	—
283	C₄×C₆₇	—	8	1	no	—
284	C₄×C₆₇	—	8	1	no	—
285	C₄×C₆₇	—	8	1	no	—
286	C₄×C₆₇	C₂×#125	1	1	yes	characteristic
287	C₄×C₆₇	#529	1	1	yes	characteristic
288	C₄×C₆₇	#529	1	1	yes	characteristic
289	C₄×C₆₇	#533	1	1	yes	characteristic
290	#33070	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
291	C₂×C₆₇	—	4	4	no	—
292	C₂×C₆₇	#2851	1	1	yes	characteristic
293	C₂×C₆₇	#3076	1	1	yes	characteristic
294	C₂×C₆₇	#3077	1	1	yes	characteristic
295	C₂×C₆₇	#3102	1	1	yes	characteristic
296	C₂×C₆₇	#3105	1	1	yes	characteristic
297	C₂×C₆₇	#3111	1	1	yes	characteristic
298	C₂×C₆₇	#3114	1	1	yes	characteristic
299	#2864	C₂×C₆₇	1	1	yes	characteristic, complemented
300	C₄²×D₄	C₂×C₆₇	1	1	yes	characteristic
301	#1949	C₂×C₆₇	1	1	yes	characteristic, complemented
302	#1949	C₂×C₆₇	1	1	yes	characteristic, complemented
303	#1921	C₂×C₆₇	1	1	yes	characteristic, complemented
304	#1920	C₂×C₆₇	1	1	yes	characteristic, complemented
305	#1919	C₂×C₆₇	1	1	yes	characteristic, complemented
306	#1918	C₂×C₆₇	1	1	yes	characteristic, complemented
307	#1864	C₂×C₆₇	1	1	yes	characteristic
308	#1864	C₂×C₆₇	1	1	yes	characteristic
309	#1862	C₂×C₆₇	1	1	yes	characteristic
310	#1862	C₂×C₆₇	1	1	yes	characteristic
311	#1831	C₂×C₆₇	1	1	yes	characteristic, complemented
312	#1804	C₂×C₆₇	1	1	yes	characteristic
313	#1804	C₂×C₆₇	1	1	yes	characteristic
314	C₆₇	#33070	1	1	yes	characteristic, direct product factor, Sylow
315	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
316	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
317	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
318	C₂²×C₄²	—	2	1	no	—
319	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
320	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
321	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
322	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
323	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
324	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
325	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
326	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
327	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
328	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
329	#389	C₂²×C₆₇	1	1	yes	characteristic
330	#389	C₂²×C₆₇	1	1	yes	characteristic
331	#389	C₂²×C₆₇	1	1	yes	characteristic
332	#389	C₂²×C₆₇	1	1	yes	characteristic
333	#387	—	2	1	no	—
334	#387	—	2	1	no	—
335	#387	—	2	1	no	—
336	#387	—	2	1	no	—
337	#382	C₂²×C₆₇	1	1	yes	characteristic
338	#382	C₂²×C₆₇	1	1	yes	characteristic
339	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
340	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
341	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
342	C₄×C₂²⋊C₄	—	2	1	no	—
343	C₄×C₂²⋊C₄	—	2	1	no	—
344	C₄×C₂²⋊C₄	—	2	1	no	—
345	C₄³	C₂²×C₆₇	1	1	yes	characteristic
346	#340	C₂²×C₆₇	1	1	yes	characteristic
347	#340	C₂²×C₆₇	1	1	yes	characteristic
348	#340	C₂²×C₆₇	1	1	yes	characteristic
349	#340	C₂²×C₆₇	1	1	yes	characteristic
350	#340	C₂²×C₆₇	1	1	yes	characteristic
351	#340	C₂²×C₆₇	1	1	yes	characteristic
352	#340	C₂²×C₆₇	1	1	yes	characteristic
353	#340	C₂²×C₆₇	1	1	yes	characteristic
354	#336	C₂²×C₆₇	1	1	yes	characteristic
355	#336	C₂²×C₆₇	1	1	yes	characteristic
356	#336	C₂²×C₆₇	1	1	yes	characteristic
357	#336	C₂²×C₆₇	1	1	yes	characteristic
358	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
359	C₂³×C₄	—	2	1	no	—
360	C₂³×C₄	—	2	1	no	—
361	C₂³×C₄	—	2	1	no	—
362	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
363	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
364	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
365	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
366	C₄×D₄	—	2	2	no	—
367	C₄×D₄	—	2	2	no	—
368	C₄×D₄	—	2	2	no	—
369	C₄×D₄	—	2	2	no	—
370	C₄×D₄	—	2	2	no	—
371	C₄×D₄	Q₈×C₆₇	1	4	yes	—
372	C₂×C₄⋊C₄	—	2	1	no	—
373	C₂×C₄⋊C₄	—	2	1	no	—
374	C₂×C₄⋊C₄	—	2	1	no	—
375	C₂×C₄⋊C₄	—	2	1	no	—
376	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
377	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
378	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
379	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
380	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
381	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
382	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
383	C₂×C₂²⋊C₄	—	2	1	no	—
384	C₂×C₂²⋊C₄	—	2	1	no	—
385	C₂×C₂²⋊C₄	—	2	1	no	—
386	C₂×C₄²	—	2	1	no	—
387	C₂×C₄²	—	2	1	no	—
388	C₂×C₄²	—	2	1	no	—
389	C₂×C₄²	—	4	4	no	—
390	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
391	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
392	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
393	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
394	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
395	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
396	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
397	#107	—	2	2	no	—
398	#107	—	2	2	no	—
399	#106	—	2	2	no	—
400	#106	—	2	2	no	—
401	C₄⋊C₈	—	2	2	no	—
402	C₄⋊C₈	—	2	2	no	—
403	#102	—	2	2	no	—
404	#102	—	2	2	no	—
405	#102	—	2	2	no	—
406	#102	—	2	2	no	—
407	#95	—	2	1	no	—
408	#95	—	2	1	no	—
409	#95	—	2	1	no	—
410	#95	—	2	1	no	—
411	#95	—	2	1	no	—
412	#95	—	2	1	no	—
413	#95	—	2	1	no	—
414	#95	—	2	1	no	—
415	#95	—	2	1	no	—
416	#95	—	2	1	no	—
417	#95	—	2	1	no	—
418	C₂⁴	—	2	1	no	—
419	C₂×D₄	—	2	1	no	—
420	C₂×D₄	—	2	1	no	—
421	C₂×D₄	—	2	1	no	—
422	C₂×D₄	C₂×Q₈×C₆₇	1	2	yes	—
423	C₂×D₄	#41×C₆₇	1	2	yes	—
424	C₂×D₄	#41×C₆₇	1	2	yes	—
425	C₂²×C₄	—	2	4	no	—
426	C₂²×C₄	—	2	4	no	—
427	C₂²×C₄	—	2	1	no	—
428	C₂²×C₄	—	2	1	no	—
429	C₂²×C₄	—	2	1	no	—
430	C₂²×C₄	—	2	4	no	—
431	C₂²×C₄	—	2	1	no	—
432	C₂²×C₄	—	2	1	no	—
433	C₂²×C₄	—	2	1	no	—
434	C₂²×C₄	—	2	1	no	—
435	C₂²×C₄	—	4	2	no	—
436	C₂²×C₄	—	4	2	no	—
437	C₂²×C₄	—	4	2	no	—
438	C₂²×C₄	—	4	2	no	—
439	C₂²×C₄	—	4	2	no	—
440	C₂²×C₄	—	4	2	no	—
441	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
442	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
443	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
444	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
445	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
446	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
447	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
448	C₂×C₈	—	2	1	no	—
449	C₂×C₈	—	2	1	no	—
450	C₂×C₈	—	2	1	no	—
451	C₂×C₈	—	2	1	no	—
452	C₂×C₈	—	2	1	no	—
453	C₂×C₈	—	2	1	no	—
454	C₂×C₈	—	2	1	no	—
455	C₂×C₈	—	2	1	no	—
456	C₂×C₈	—	2	1	no	—
457	C₂×C₈	—	2	1	no	—
458	C₂×C₈	—	2	1	no	—
459	C₂×C₈	—	2	1	no	—
460	C₄⋊C₄	—	2	1	no	—
461	C₄⋊C₄	—	2	1	no	—
462	C₄⋊C₄	—	2	1	no	—
463	C₄⋊C₄	—	2	1	no	—
464	C₄⋊C₄	—	2	1	no	—
465	C₄⋊C₄	—	2	1	no	—
466	C₄⋊C₄	—	2	1	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	2	1	no	—
471	C₄⋊C₄	—	4	2	no	—
472	C₄⋊C₄	—	4	2	no	—
473	C₄⋊C₄	—	4	2	no	—
474	C₄⋊C₄	—	4	2	no	—
475	C₄⋊C₄	C₂×Q₈×C₆₇	1	2	yes	—
476	C₄⋊C₄	#41×C₆₇	1	2	yes	—
477	C₄⋊C₄	#41×C₆₇	1	2	yes	—
478	C₂²⋊C₄	—	2	4	no	—
479	C₂²⋊C₄	—	2	4	no	—
480	C₂²⋊C₄	—	2	4	no	—
481	C₄²	—	2	2	no	—
482	C₄²	—	2	2	no	—
483	C₄²	—	2	1	no	—
484	C₄²	—	2	1	no	—
485	C₄²	—	2	1	no	—
486	C₄²	—	2	1	no	—
487	C₄²	—	4	1	no	—
488	C₄²	—	4	1	no	—
489	C₄²	—	4	2	no	—
490	C₄²	—	4	2	no	—
491	C₄²	—	4	2	no	—
492	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
493	C₄²	#41×C₆₇	1	2	yes	—
494	C₂³	—	2	2	no	—
495	C₂³	—	2	2	no	—
496	C₂³	—	2	2	no	—
497	C₂³	—	4	1	no	—
498	C₂³	—	4	1	no	—
499	C₂³	—	4	1	no	—
500	C₂³	—	4	1	no	—
501	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
502	D₄	—	2	2	no	—
503	D₄	—	2	2	no	—
504	D₄	—	2	2	no	—
505	D₄	#125×C₆₇	1	4	yes	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	1	no	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	4	8	no	—
524	C₂×C₄	—	4	8	no	—
525	C₂×C₄	—	4	8	no	—
526	C₂×C₄	—	4	1	no	—
527	C₂×C₄	—	4	1	no	—
528	C₂×C₄	—	4	1	no	—
529	C₂×C₄	—	4	1	no	—
530	C₂×C₄	—	4	1	no	—
531	C₂×C₄	—	4	1	no	—
532	C₂×C₄	—	4	1	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	4	1	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	4	1	no	—
540	C₂×C₄	—	4	1	no	—
541	C₂×C₄	—	4	1	no	—
542	C₂×C₄	—	4	1	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
545	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
546	C₂×C₄	#135×C₆₇	1	2	yes	—
547	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
548	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
549	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
550	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
551	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
552	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
553	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
554	C₈	—	4	1	no	—
555	C₈	—	4	1	no	—
556	C₈	—	4	1	no	—
557	C₈	—	4	1	no	—
558	C₂²	—	2	4	no	—
559	C₂²	—	4	2	no	—
560	C₂²	—	4	2	no	—
561	C₂²	—	4	2	no	—
562	C₂²	—	4	2	no	—
563	C₂²	—	4	2	no	—
564	C₂²	—	4	2	no	—
565	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
566	C₂²	#525×C₆₇	1	1	yes	characteristic
567	C₂²	#525×C₆₇	1	1	yes	characteristic
568	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
569	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
570	C₂²	#575×C₆₇	1	1	yes	characteristic
571	C₂²	#575×C₆₇	1	1	yes	characteristic
572	C₄	—	2	2	no	—
573	C₄	—	2	2	no	—
574	C₄	—	2	2	no	—
575	C₄	—	2	2	no	—
576	C₄	—	4	2	no	—
577	C₄	—	4	1	no	—
578	C₄	—	4	2	no	—
579	C₄	—	4	1	no	—
580	C₄	—	4	2	no	—
581	C₄	—	8	1	no	—
582	C₄	—	8	1	no	—
583	C₄	—	8	1	no	—
584	C₄	—	8	1	no	—
585	C₄	C₂×#125×C₆₇	1	1	yes	characteristic
586	C₄	#529×C₆₇	1	1	yes	characteristic
587	C₄	#529×C₆₇	1	1	yes	characteristic
588	C₄	#533×C₆₇	1	1	yes	characteristic
589	C₂	—	4	4	no	—
590	C₂	C₆₇×#2851	1	1	yes	characteristic
591	C₂	C₆₇×#3076	1	1	yes	characteristic
592	C₂	C₆₇×#3077	1	1	yes	characteristic
593	C₂	C₆₇×#3102	1	1	yes	characteristic
594	C₂	C₆₇×#3105	1	1	yes	characteristic
595	C₂	C₆₇×#3111	1	1	yes	characteristic
596	C₂	C₆₇×#3114	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×D₄×C₆₇	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#2864	C₂	semidirect
3. C₆₇×#1949	C₂	semidirect
4. C₆₇×#1949	C₂	semidirect
5. C₆₇×#1921	C₂	semidirect
6. C₆₇×#1920	C₂	semidirect
7. C₆₇×#1919	C₂	semidirect
8. C₆₇×#1918	C₂	semidirect
13. C₆₇×#1831	C₂	semidirect
290. #33070	C₆₇	direct (N × H)
299. #2864	C₂×C₆₇	semidirect
301. #1949	C₂×C₆₇	semidirect
302. #1949	C₂×C₆₇	semidirect
303. #1921	C₂×C₆₇	semidirect
304. #1920	C₂×C₆₇	semidirect
305. #1919	C₂×C₆₇	semidirect
306. #1918	C₂×C₆₇	semidirect
311. #1831	C₂×C₆₇	semidirect
314. C₆₇	#33070	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#2864	2
C₄²×D₄×C₆₇	2
C₆₇×#1949	2
C₆₇×#1921	2
C₆₇×#1920	2
C₆₇×#1919	2
C₆₇×#1918	2
C₆₇×#1864	2
C₆₇×#1862	2
C₆₇×#1831	2
C₆₇×#1804	2
#33070	67

Subgroups — C67×#33070 #30712321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33070 #30712321