Subgroups — C₆₇×#21814 #30701065

Order 17152.

1250 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#536×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#433×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2314	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1983	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1981	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1960	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1959	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1958	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1956	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1788	C₂	1	2	yes	maximal
11	C₆₇×#1787	C₂	1	2	yes	maximal, complemented
12	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
13	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	#536×C₆₇	—	2	4	no	—
15	#536×C₆₇	C₄	1	8	yes	complemented
16	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#123×C₆₇	—	2	2	no	—
21	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
22	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
23	#433×C₆₇	—	2	4	no	—
24	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
25	C₂×#103×C₆₇	C₂²	1	2	yes	—
26	C₂×#103×C₆₇	C₂²	1	2	yes	—
27	C₂×#103×C₆₇	C₂²	1	2	yes	—
28	C₂×#103×C₆₇	C₂²	1	2	yes	—
29	C₂×#103×C₆₇	C₂²	1	2	yes	—
30	C₂×#103×C₆₇	C₂²	1	2	yes	—
31	C₂×#102×C₆₇	C₂²	1	2	yes	—
32	C₂×#102×C₆₇	C₂²	1	2	yes	—
33	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
34	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
36	#388×C₆₇	—	2	2	no	—
37	#387×C₆₇	—	2	2	no	—
38	#384×C₆₇	C₂²	1	1	yes	characteristic
39	#382×C₆₇	C₂²	1	1	yes	characteristic
40	#380×C₆₇	C₂²	1	2	yes	—
41	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
42	C₂×#41×C₆₇	—	2	2	no	—
43	C₂×#41×C₆₇	—	4	2	no	—
44	C₂×#41×C₆₇	—	4	2	no	—
45	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
46	C₂²×Q₈×C₆₇	—	2	1	no	—
47	C₂²×Q₈×C₆₇	—	2	2	no	—
48	C₂²×Q₈×C₆₇	—	4	2	no	—
49	C₂²×Q₈×C₆₇	—	4	2	no	—
50	C₂²×Q₈×C₆₇	C₂×C₄	1	4	yes	—
51	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
54	C₂²×D₄×C₆₇	—	2	1	no	—
55	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
56	C₂³×C₄×C₆₇	—	2	1	no	—
57	C₂³×C₄×C₆₇	—	2	1	no	—
58	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
60	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
61	C₄⋊Q₈×C₆₇	—	2	2	no	—
62	C₄⋊Q₈×C₆₇	—	2	2	no	—
63	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
64	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
65	#124×C₆₇	—	2	2	no	—
66	#124×C₆₇	—	2	2	no	—
67	#124×C₆₇	C₂×C₄	1	4	yes	—
68	#124×C₆₇	C₂×C₄	1	4	yes	—
69	#123×C₆₇	—	2	16	no	—
70	C₂²⋊Q₈×C₆₇	—	2	16	no	—
71	#117×C₆₇	C₂×C₄	1	4	yes	—
72	#117×C₆₇	D₄	1	4	yes	—
73	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
74	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
75	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
76	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
77	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
78	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
79	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
80	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
81	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
82	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
83	C₂×C₄²×C₆₇	—	2	2	no	—
84	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
85	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
86	C₄⋊C₈×C₆₇	—	2	4	no	—
87	#103×C₆₇	—	4	2	no	—
88	#103×C₆₇	—	4	2	no	—
89	#103×C₆₇	—	4	2	no	—
90	#103×C₆₇	—	4	2	no	—
91	#103×C₆₇	—	4	2	no	—
92	#103×C₆₇	—	4	2	no	—
93	#102×C₆₇	—	4	2	no	—
94	#102×C₆₇	—	4	2	no	—
95	C₂²⋊C₈×C₆₇	—	2	4	no	—
96	#97×C₆₇	—	2	2	no	—
97	C₄×C₈×C₆₇	—	2	2	no	—
98	#95×C₆₇	—	2	2	no	—
99	#95×C₆₇	—	2	2	no	—
100	#95×C₆₇	—	2	2	no	—
101	C₂⁴×C₆₇	—	2	1	no	—
102	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
103	#41×C₆₇	—	4	8	no	—
104	#41×C₆₇	—	4	8	no	—
105	C₂×Q₈×C₆₇	—	2	4	no	—
106	C₂×Q₈×C₆₇	—	2	1	no	—
107	C₂×Q₈×C₆₇	—	2	1	no	—
108	C₂×Q₈×C₆₇	—	2	1	no	—
109	C₂×Q₈×C₆₇	—	2	2	no	—
110	C₂×Q₈×C₆₇	—	2	2	no	—
111	C₂×Q₈×C₆₇	—	2	2	no	—
112	C₂×Q₈×C₆₇	—	2	2	no	—
113	C₂×Q₈×C₆₇	—	2	2	no	—
114	C₂×Q₈×C₆₇	—	2	2	no	—
115	C₂×Q₈×C₆₇	—	4	8	no	—
116	C₂×Q₈×C₆₇	—	4	2	no	—
117	C₂×Q₈×C₆₇	—	4	2	no	—
118	C₂×Q₈×C₆₇	—	4	8	no	—
119	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
120	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
121	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
122	C₂×D₄×C₆₇	—	2	4	no	—
123	C₂×D₄×C₆₇	—	2	1	no	—
124	C₂×D₄×C₆₇	—	2	2	no	—
125	C₂×D₄×C₆₇	—	2	2	no	—
126	C₂×D₄×C₆₇	—	4	2	no	—
127	C₂×D₄×C₆₇	—	4	2	no	—
128	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
129	C₂²×C₄×C₆₇	—	2	2	no	—
130	C₂²×C₄×C₆₇	—	2	2	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	4	no	—
136	C₂²×C₄×C₆₇	—	2	2	no	—
137	C₂²×C₄×C₆₇	—	2	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	C₂²×C₄	1	2	yes	—
143	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
144	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
145	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
147	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
148	C₂×C₈×C₆₇	—	2	2	no	—
149	C₂×C₈×C₆₇	—	2	2	no	—
150	C₂×C₈×C₆₇	—	2	2	no	—
151	C₂×C₈×C₆₇	—	2	2	no	—
152	C₂×C₈×C₆₇	—	2	2	no	—
153	C₂×C₈×C₆₇	—	2	2	no	—
154	C₄⋊C₄×C₆₇	—	2	8	no	—
155	C₄⋊C₄×C₆₇	—	2	8	no	—
156	C₄⋊C₄×C₆₇	—	2	2	no	—
157	C₄⋊C₄×C₆₇	—	2	2	no	—
158	C₄⋊C₄×C₆₇	—	2	2	no	—
159	C₄⋊C₄×C₆₇	—	2	2	no	—
160	C₄⋊C₄×C₆₇	—	8	2	no	—
161	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
162	C₂²⋊C₄×C₆₇	—	2	8	no	—
163	C₂²⋊C₄×C₆₇	—	2	8	no	—
164	C₂²⋊C₄×C₆₇	—	8	2	no	—
165	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
166	C₄²×C₆₇	—	8	2	no	—
167	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
168	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
169	C₄²×C₆₇	C₂×D₄	1	2	yes	—
170	C₄²×C₆₇	C₂×D₄	1	2	yes	—
171	C₂³×C₆₇	—	2	1	no	—
172	C₂³×C₆₇	—	2	1	no	—
173	C₂³×C₆₇	—	2	2	no	—
174	C₂³×C₆₇	—	2	2	no	—
175	C₂³×C₆₇	—	2	2	no	—
176	C₂³×C₆₇	—	4	1	no	—
177	C₂³×C₆₇	—	4	1	no	—
178	C₂³×C₆₇	—	4	2	no	—
179	C₂³×C₆₇	—	4	2	no	—
180	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
181	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
182	Q₈×C₆₇	—	2	4	no	—
183	Q₈×C₆₇	—	2	4	no	—
184	Q₈×C₆₇	—	2	4	no	—
185	Q₈×C₆₇	—	4	8	no	—
186	Q₈×C₆₇	—	4	2	no	—
187	Q₈×C₆₇	—	4	2	no	—
188	Q₈×C₆₇	—	4	2	no	—
189	D₄×C₆₇	—	2	4	no	—
190	D₄×C₆₇	—	4	8	no	—
191	D₄×C₆₇	—	4	2	no	—
192	C₂×C₄×C₆₇	—	2	1	no	—
193	C₂×C₄×C₆₇	—	2	1	no	—
194	C₂×C₄×C₆₇	—	2	2	no	—
195	C₂×C₄×C₆₇	—	2	2	no	—
196	C₂×C₄×C₆₇	—	2	2	no	—
197	C₂×C₄×C₆₇	—	2	2	no	—
198	C₂×C₄×C₆₇	—	2	4	no	—
199	C₂×C₄×C₆₇	—	2	4	no	—
200	C₂×C₄×C₆₇	—	2	2	no	—
201	C₂×C₄×C₆₇	—	2	2	no	—
202	C₂×C₄×C₆₇	—	2	2	no	—
203	C₂×C₄×C₆₇	—	4	8	no	—
204	C₂×C₄×C₆₇	—	4	8	no	—
205	C₂×C₄×C₆₇	—	4	1	no	—
206	C₂×C₄×C₆₇	—	4	1	no	—
207	C₂×C₄×C₆₇	—	4	1	no	—
208	C₂×C₄×C₆₇	—	4	1	no	—
209	C₂×C₄×C₆₇	—	4	1	no	—
210	C₂×C₄×C₆₇	—	4	1	no	—
211	C₂×C₄×C₆₇	—	4	2	no	—
212	C₂×C₄×C₆₇	—	4	2	no	—
213	C₂×C₄×C₆₇	—	4	2	no	—
214	C₂×C₄×C₆₇	—	4	2	no	—
215	C₂×C₄×C₆₇	—	4	2	no	—
216	C₂×C₄×C₆₇	—	4	2	no	—
217	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
218	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
219	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
220	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
221	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
223	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
224	C₈×C₆₇	—	4	2	no	—
225	C₈×C₆₇	—	4	2	no	—
226	C₂²×C₆₇	—	2	2	no	—
227	C₂²×C₆₇	—	2	4	no	—
228	C₂²×C₆₇	—	4	1	no	—
229	C₂²×C₆₇	—	4	1	no	—
230	C₂²×C₆₇	—	4	2	no	—
231	C₂²×C₆₇	—	4	2	no	—
232	C₂²×C₆₇	—	4	4	no	—
233	C₂²×C₆₇	—	8	1	no	—
234	C₂²×C₆₇	—	8	1	no	—
235	C₂²×C₆₇	—	8	2	no	—
236	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
237	C₂²×C₆₇	#535	1	1	yes	characteristic
238	C₂²×C₆₇	#535	1	1	yes	characteristic
239	C₂²×C₆₇	#577	1	2	yes	—
240	C₂²×C₆₇	#578	1	2	yes	—
241	C₄×C₆₇	—	2	2	no	—
242	C₄×C₆₇	—	2	4	no	—
243	C₄×C₆₇	—	2	4	no	—
244	C₄×C₆₇	—	4	2	no	—
245	C₄×C₆₇	—	4	2	no	—
246	C₄×C₆₇	—	4	2	no	—
247	C₄×C₆₇	—	8	2	no	—
248	C₄×C₆₇	—	8	2	no	—
249	C₄×C₆₇	#518	1	2	yes	—
250	C₄×C₆₇	#535	1	2	yes	—
251	#21814	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
252	C₂×C₆₇	—	4	2	no	—
253	C₂×C₆₇	—	8	2	no	—
254	C₂×C₆₇	#2309	1	1	yes	characteristic
255	C₂×C₆₇	#2898	1	2	yes	—
256	C₂×C₆₇	#3221	1	1	yes	characteristic
257	C₂×C₆₇	#3223	1	1	yes	characteristic
258	C₂×C₆₇	#3225	1	2	yes	—
259	C₂×#536	C₂×C₆₇	1	1	yes	characteristic
260	C₂×#433	C₂×C₆₇	1	1	yes	characteristic, complemented
261	#2314	C₂×C₆₇	1	1	yes	characteristic, complemented
262	#1983	C₂×C₆₇	1	2	yes	complemented
263	#1981	C₂×C₆₇	1	2	yes	complemented
264	#1960	C₂×C₆₇	1	1	yes	characteristic, complemented
265	#1959	C₂×C₆₇	1	1	yes	characteristic, complemented
266	#1958	C₂×C₆₇	1	1	yes	characteristic, complemented
267	#1956	C₂×C₆₇	1	1	yes	characteristic, complemented
268	#1788	C₂×C₆₇	1	2	yes	—
269	#1787	C₂×C₆₇	1	2	yes	complemented
270	C₆₇	#21814	1	1	yes	characteristic, direct product factor, Sylow
271	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
272	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
273	#536	—	2	4	no	—
274	#536	C₄×C₆₇	1	8	yes	complemented
275	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
276	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
277	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
278	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
279	C₂×#123	—	2	2	no	—
280	C₂×C₂²⋊Q₈	—	2	2	no	—
281	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
282	#433	—	2	4	no	—
283	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
284	C₂×#103	C₂²×C₆₇	1	2	yes	—
285	C₂×#103	C₂²×C₆₇	1	2	yes	—
286	C₂×#103	C₂²×C₆₇	1	2	yes	—
287	C₂×#103	C₂²×C₆₇	1	2	yes	—
288	C₂×#103	C₂²×C₆₇	1	2	yes	—
289	C₂×#103	C₂²×C₆₇	1	2	yes	—
290	C₂×#102	C₂²×C₆₇	1	2	yes	—
291	C₂×#102	C₂²×C₆₇	1	2	yes	—
292	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
293	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
294	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
295	#388	—	2	2	no	—
296	#387	—	2	2	no	—
297	#384	C₂²×C₆₇	1	1	yes	characteristic
298	#382	C₂²×C₆₇	1	1	yes	characteristic
299	#380	C₂²×C₆₇	1	2	yes	—
300	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
301	C₂×#41	—	2	2	no	—
302	C₂×#41	—	4	2	no	—
303	C₂×#41	—	4	2	no	—
304	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
305	C₂²×Q₈	—	2	1	no	—
306	C₂²×Q₈	—	2	2	no	—
307	C₂²×Q₈	—	4	2	no	—
308	C₂²×Q₈	—	4	2	no	—
309	C₂²×Q₈	C₂×C₄×C₆₇	1	4	yes	—
310	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
311	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
312	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
313	C₂²×D₄	—	2	1	no	—
314	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
315	C₂³×C₄	—	2	1	no	—
316	C₂³×C₄	—	2	1	no	—
317	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
318	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
319	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
320	C₄⋊Q₈	—	2	2	no	—
321	C₄⋊Q₈	—	2	2	no	—
322	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
323	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
324	#124	—	2	2	no	—
325	#124	—	2	2	no	—
326	#124	C₂×C₄×C₆₇	1	4	yes	—
327	#124	C₂×C₄×C₆₇	1	4	yes	—
328	#123	—	2	16	no	—
329	C₂²⋊Q₈	—	2	16	no	—
330	#117	C₂×C₄×C₆₇	1	4	yes	—
331	#117	D₄×C₆₇	1	4	yes	—
332	C₂×C₄⋊C₄	—	2	2	no	—
333	C₂×C₄⋊C₄	—	2	2	no	—
334	C₂×C₄⋊C₄	—	2	2	no	—
335	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
336	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
337	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
338	C₂×C₂²⋊C₄	—	2	2	no	—
339	C₂×C₂²⋊C₄	—	2	2	no	—
340	C₂×C₂²⋊C₄	—	2	2	no	—
341	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
342	C₂×C₄²	—	2	2	no	—
343	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
344	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
345	C₄⋊C₈	—	2	4	no	—
346	#103	—	4	2	no	—
347	#103	—	4	2	no	—
348	#103	—	4	2	no	—
349	#103	—	4	2	no	—
350	#103	—	4	2	no	—
351	#103	—	4	2	no	—
352	#102	—	4	2	no	—
353	#102	—	4	2	no	—
354	C₂²⋊C₈	—	2	4	no	—
355	#97	—	2	2	no	—
356	C₄×C₈	—	2	2	no	—
357	#95	—	2	2	no	—
358	#95	—	2	2	no	—
359	#95	—	2	2	no	—
360	C₂⁴	—	2	1	no	—
361	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
362	#41	—	4	8	no	—
363	#41	—	4	8	no	—
364	C₂×Q₈	—	2	4	no	—
365	C₂×Q₈	—	2	1	no	—
366	C₂×Q₈	—	2	1	no	—
367	C₂×Q₈	—	2	1	no	—
368	C₂×Q₈	—	2	2	no	—
369	C₂×Q₈	—	2	2	no	—
370	C₂×Q₈	—	2	2	no	—
371	C₂×Q₈	—	2	2	no	—
372	C₂×Q₈	—	2	2	no	—
373	C₂×Q₈	—	2	2	no	—
374	C₂×Q₈	—	4	8	no	—
375	C₂×Q₈	—	4	2	no	—
376	C₂×Q₈	—	4	2	no	—
377	C₂×Q₈	—	4	8	no	—
378	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
379	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
380	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
381	C₂×D₄	—	2	4	no	—
382	C₂×D₄	—	2	1	no	—
383	C₂×D₄	—	2	2	no	—
384	C₂×D₄	—	2	2	no	—
385	C₂×D₄	—	4	2	no	—
386	C₂×D₄	—	4	2	no	—
387	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	4	no	—
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	2	1	no	—
393	C₂²×C₄	—	2	1	no	—
394	C₂²×C₄	—	2	4	no	—
395	C₂²×C₄	—	2	2	no	—
396	C₂²×C₄	—	2	2	no	—
397	C₂²×C₄	—	4	2	no	—
398	C₂²×C₄	—	4	2	no	—
399	C₂²×C₄	—	4	2	no	—
400	C₂²×C₄	—	4	2	no	—
401	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
402	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
403	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
404	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
405	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
406	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
407	C₂×C₈	—	2	2	no	—
408	C₂×C₈	—	2	2	no	—
409	C₂×C₈	—	2	2	no	—
410	C₂×C₈	—	2	2	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	2	2	no	—
413	C₄⋊C₄	—	2	8	no	—
414	C₄⋊C₄	—	2	8	no	—
415	C₄⋊C₄	—	2	2	no	—
416	C₄⋊C₄	—	2	2	no	—
417	C₄⋊C₄	—	2	2	no	—
418	C₄⋊C₄	—	2	2	no	—
419	C₄⋊C₄	—	8	2	no	—
420	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
421	C₂²⋊C₄	—	2	8	no	—
422	C₂²⋊C₄	—	2	8	no	—
423	C₂²⋊C₄	—	8	2	no	—
424	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
425	C₄²	—	8	2	no	—
426	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
427	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
428	C₄²	C₂×D₄×C₆₇	1	2	yes	—
429	C₄²	C₂×D₄×C₆₇	1	2	yes	—
430	C₂³	—	2	1	no	—
431	C₂³	—	2	1	no	—
432	C₂³	—	2	2	no	—
433	C₂³	—	2	2	no	—
434	C₂³	—	2	2	no	—
435	C₂³	—	4	1	no	—
436	C₂³	—	4	1	no	—
437	C₂³	—	4	2	no	—
438	C₂³	—	4	2	no	—
439	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
440	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
441	Q₈	—	2	4	no	—
442	Q₈	—	2	4	no	—
443	Q₈	—	2	4	no	—
444	Q₈	—	4	8	no	—
445	Q₈	—	4	2	no	—
446	Q₈	—	4	2	no	—
447	Q₈	—	4	2	no	—
448	D₄	—	2	4	no	—
449	D₄	—	4	8	no	—
450	D₄	—	4	2	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	4	8	no	—
463	C₂×C₄	—	4	8	no	—
464	C₂×C₄	—	4	1	no	—
465	C₂×C₄	—	4	1	no	—
466	C₂×C₄	—	4	1	no	—
467	C₂×C₄	—	4	1	no	—
468	C₂×C₄	—	4	1	no	—
469	C₂×C₄	—	4	1	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	—	4	2	no	—
472	C₂×C₄	—	4	2	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
477	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
478	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
479	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
480	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
481	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
482	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
483	C₈	—	4	2	no	—
484	C₈	—	4	2	no	—
485	C₂²	—	2	2	no	—
486	C₂²	—	2	4	no	—
487	C₂²	—	4	1	no	—
488	C₂²	—	4	1	no	—
489	C₂²	—	4	2	no	—
490	C₂²	—	4	2	no	—
491	C₂²	—	4	4	no	—
492	C₂²	—	8	1	no	—
493	C₂²	—	8	1	no	—
494	C₂²	—	8	2	no	—
495	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
496	C₂²	#535×C₆₇	1	1	yes	characteristic
497	C₂²	#535×C₆₇	1	1	yes	characteristic
498	C₂²	#577×C₆₇	1	2	yes	—
499	C₂²	#578×C₆₇	1	2	yes	—
500	C₄	—	2	2	no	—
501	C₄	—	2	4	no	—
502	C₄	—	2	4	no	—
503	C₄	—	4	2	no	—
504	C₄	—	4	2	no	—
505	C₄	—	4	2	no	—
506	C₄	—	8	2	no	—
507	C₄	—	8	2	no	—
508	C₄	#518×C₆₇	1	2	yes	—
509	C₄	#535×C₆₇	1	2	yes	—
510	C₂	—	4	2	no	—
511	C₂	—	8	2	no	—
512	C₂	C₆₇×#2309	1	1	yes	characteristic
513	C₂	C₆₇×#2898	1	2	yes	—
514	C₂	C₆₇×#3221	1	1	yes	characteristic
515	C₂	C₆₇×#3223	1	1	yes	characteristic
516	C₂	C₆₇×#3225	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#433×C₆₇	C₂	semidirect
3. C₆₇×#2314	C₂	semidirect
4. C₆₇×#1983	C₂	semidirect
5. C₆₇×#1981	C₂	semidirect
6. C₆₇×#1960	C₂	semidirect
7. C₆₇×#1959	C₂	semidirect
8. C₆₇×#1958	C₂	semidirect
9. C₆₇×#1956	C₂	semidirect
11. C₆₇×#1787	C₂	semidirect
15. #536×C₆₇	C₄	semidirect
251. #21814	C₆₇	direct (N × H)
260. C₂×#433	C₂×C₆₇	semidirect
261. #2314	C₂×C₆₇	semidirect
262. #1983	C₂×C₆₇	semidirect
263. #1981	C₂×C₆₇	semidirect
264. #1960	C₂×C₆₇	semidirect
265. #1959	C₂×C₆₇	semidirect
266. #1958	C₂×C₆₇	semidirect
267. #1956	C₂×C₆₇	semidirect
269. #1787	C₂×C₆₇	semidirect
270. C₆₇	#21814	direct (N × H)
274. #536	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#536×C₆₇	2
C₂×#433×C₆₇	2
C₆₇×#2314	2
C₆₇×#1983	2
C₆₇×#1981	2
C₆₇×#1960	2
C₆₇×#1959	2
C₆₇×#1958	2
C₆₇×#1956	2
C₆₇×#1788	2
C₆₇×#1787	2
#21814	67

Subgroups — C67×#21814 #30701065

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21814 #30701065