Subgroups — C₆₇×#36699 #30715950

Order 17152.

1958 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#533×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#529×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2822	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2821	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2821	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2820	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2817	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2816	C₂	1	2	yes	maximal
9	C₆₇×#2814	C₂	1	4	yes	maximal, complemented
10	C₆₇×#2399	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2398	C₂	1	2	yes	maximal, complemented
12	C₆₇×#2396	C₂	1	4	yes	maximal, complemented
13	C₆₇×#2395	C₂	1	4	yes	maximal, complemented
14	C₆₇×#2283	C₂	1	1	yes	maximal, characteristic, complemented
15	#533×C₆₇	—	2	8	no	—
16	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
18	#529×C₆₇	—	2	8	no	—
19	C₂×#126×C₆₇	C₂²	1	2	yes	—
20	C₂×#125×C₆₇	C₂²	1	2	yes	—
21	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#123×C₆₇	C₂²	1	2	yes	—
24	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
25	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
26	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
29	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
30	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
32	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×#117×C₆₇	C₂²	1	2	yes	—
34	C₂×#117×C₆₇	C₂²	1	2	yes	—
35	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
36	#400×C₆₇	C₂²	1	4	yes	—
37	#400×C₆₇	C₂²	1	4	yes	—
38	#400×C₆₇	C₂²	1	4	yes	—
39	#399×C₆₇	C₂²	1	4	yes	—
40	#399×C₆₇	C₂²	1	4	yes	—
41	#398×C₆₇	C₂²	1	4	yes	—
42	#398×C₆₇	C₂²	1	8	yes	—
43	#398×C₆₇	C₂²	1	4	yes	—
44	#398×C₆₇	C₂²	1	4	yes	—
45	#397×C₆₇	C₂²	1	8	yes	—
46	#397×C₆₇	C₂²	1	4	yes	—
47	#397×C₆₇	C₂²	1	4	yes	—
48	#396×C₆₇	C₂²	1	8	yes	—
49	#395×C₆₇	C₂²	1	4	yes	—
50	#395×C₆₇	C₂²	1	4	yes	—
51	#395×C₆₇	C₂²	1	4	yes	—
52	#394×C₆₇	C₂²	1	8	yes	—
53	#394×C₆₇	C₂²	1	4	yes	—
54	#393×C₆₇	C₂²	1	4	yes	—
55	#393×C₆₇	C₂²	1	4	yes	—
56	#383×C₆₇	C₂²	1	2	yes	—
57	#382×C₆₇	C₂²	1	2	yes	—
58	#381×C₆₇	C₂²	1	2	yes	—
59	#380×C₆₇	C₂²	1	2	yes	—
60	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
61	C₄×C₄⋊C₄×C₆₇	C₂²	1	8	yes	—
62	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
63	C₄×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	—
64	#376×C₆₇	C₂²	1	2	yes	—
65	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
66	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
67	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
68	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
69	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
71	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
74	C₄⋊Q₈×C₆₇	—	2	4	no	—
75	C₄⋊Q₈×C₆₇	—	2	4	no	—
76	#126×C₆₇	—	2	8	no	—
77	#125×C₆₇	—	2	4	no	—
78	#125×C₆₇	—	2	8	no	—
79	#124×C₆₇	—	2	4	no	—
80	#123×C₆₇	—	2	8	no	—
81	C₂²⋊Q₈×C₆₇	—	2	4	no	—
82	C₂²⋊Q₈×C₆₇	—	2	8	no	—
83	C₂²⋊Q₈×C₆₇	—	2	8	no	—
84	#121×C₆₇	—	2	4	no	—
85	C₄×Q₈×C₆₇	—	2	4	no	—
86	C₄×Q₈×C₆₇	—	2	8	no	—
87	C₄×Q₈×C₆₇	—	2	8	no	—
88	C₄×D₄×C₆₇	—	2	8	no	—
89	C₄×D₄×C₆₇	—	2	4	no	—
90	#117×C₆₇	—	2	8	no	—
91	#117×C₆₇	—	2	8	no	—
92	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
93	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
94	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
95	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
97	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
98	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
100	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
101	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
102	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
105	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
106	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
107	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
108	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
109	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
110	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
111	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
112	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
113	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
114	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
115	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
116	C₂×C₄²×C₆₇	D₄	1	8	yes	—
117	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
118	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
119	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
120	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
122	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
123	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
124	#95×C₆₇	C₂³	1	4	yes	—
125	#95×C₆₇	C₂³	1	4	yes	—
126	#95×C₆₇	C₂³	1	4	yes	—
127	#95×C₆₇	C₂³	1	8	yes	—
128	#95×C₆₇	C₂³	1	8	yes	—
129	#95×C₆₇	C₂³	1	4	yes	—
130	#95×C₆₇	C₂³	1	8	yes	—
131	#95×C₆₇	C₂³	1	8	yes	—
132	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
133	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
134	C₂×Q₈×C₆₇	—	2	2	no	—
135	C₂×Q₈×C₆₇	—	2	4	no	—
136	C₂×Q₈×C₆₇	—	2	4	no	—
137	C₂×Q₈×C₆₇	—	4	2	no	—
138	C₂×Q₈×C₆₇	—	4	4	no	—
139	C₂×Q₈×C₆₇	—	4	4	no	—
140	C₂×D₄×C₆₇	—	2	4	no	—
141	C₂×D₄×C₆₇	—	2	2	no	—
142	C₂×D₄×C₆₇	—	4	2	no	—
143	C₂×D₄×C₆₇	—	4	4	no	—
144	C₂²×C₄×C₆₇	—	2	4	no	—
145	C₂²×C₄×C₆₇	—	2	4	no	—
146	C₂²×C₄×C₆₇	—	2	8	no	—
147	C₂²×C₄×C₆₇	—	4	2	no	—
148	C₂²×C₄×C₆₇	—	4	2	no	—
149	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
150	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
151	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
152	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
153	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
154	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
155	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
156	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
157	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
158	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
159	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
160	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
161	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
162	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
163	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
164	C₄⋊C₄×C₆₇	—	2	4	no	—
165	C₄⋊C₄×C₆₇	—	2	4	no	—
166	C₄⋊C₄×C₆₇	—	2	4	no	—
167	C₄⋊C₄×C₆₇	—	2	4	no	—
168	C₄⋊C₄×C₆₇	—	2	2	no	—
169	C₄⋊C₄×C₆₇	—	2	4	no	—
170	C₄⋊C₄×C₆₇	—	2	2	no	—
171	C₄⋊C₄×C₆₇	—	2	8	no	—
172	C₄⋊C₄×C₆₇	—	2	8	no	—
173	C₄⋊C₄×C₆₇	—	2	4	no	—
174	C₄⋊C₄×C₆₇	—	2	4	no	—
175	C₄⋊C₄×C₆₇	—	2	4	no	—
176	C₄⋊C₄×C₆₇	—	4	4	no	—
177	C₄⋊C₄×C₆₇	—	4	8	no	—
178	C₄⋊C₄×C₆₇	—	4	8	no	—
179	C₂²⋊C₄×C₆₇	—	2	8	no	—
180	C₂²⋊C₄×C₆₇	—	2	4	no	—
181	C₂²⋊C₄×C₆₇	—	2	4	no	—
182	C₂²⋊C₄×C₆₇	—	2	4	no	—
183	C₂²⋊C₄×C₆₇	—	2	4	no	—
184	C₂²⋊C₄×C₆₇	—	2	4	no	—
185	C₂²⋊C₄×C₆₇	—	4	4	no	—
186	C₂²⋊C₄×C₆₇	—	4	8	no	—
187	C₂²⋊C₄×C₆₇	—	4	8	no	—
188	C₄²×C₆₇	—	2	4	no	—
189	C₄²×C₆₇	—	2	8	no	—
190	C₄²×C₆₇	—	2	8	no	—
191	C₄²×C₆₇	C₂×D₄	1	4	yes	—
192	C₄²×C₆₇	C₂×D₄	1	4	yes	—
193	C₄²×C₆₇	C₂×D₄	1	4	yes	—
194	C₄²×C₆₇	C₂×D₄	1	4	yes	—
195	C₂³×C₆₇	—	2	2	no	—
196	C₂³×C₆₇	—	2	2	no	—
197	C₂³×C₆₇	—	2	2	no	—
198	C₂³×C₆₇	—	2	2	no	—
199	C₂³×C₆₇	—	2	4	no	—
200	C₂³×C₆₇	—	2	4	no	—
201	C₂³×C₆₇	—	2	4	no	—
202	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
203	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
204	Q₈×C₆₇	—	4	4	no	—
205	Q₈×C₆₇	—	4	8	no	—
206	Q₈×C₆₇	—	4	8	no	—
207	D₄×C₆₇	—	4	4	no	—
208	D₄×C₆₇	—	4	8	no	—
209	C₂×C₄×C₆₇	—	2	2	no	—
210	C₂×C₄×C₆₇	—	2	2	no	—
211	C₂×C₄×C₆₇	—	2	2	no	—
212	C₂×C₄×C₆₇	—	2	2	no	—
213	C₂×C₄×C₆₇	—	2	2	no	—
214	C₂×C₄×C₆₇	—	2	8	no	—
215	C₂×C₄×C₆₇	—	2	2	no	—
216	C₂×C₄×C₆₇	—	2	2	no	—
217	C₂×C₄×C₆₇	—	2	4	no	—
218	C₂×C₄×C₆₇	—	2	4	no	—
219	C₂×C₄×C₆₇	—	2	2	no	—
220	C₂×C₄×C₆₇	—	2	8	no	—
221	C₂×C₄×C₆₇	—	2	4	no	—
222	C₂×C₄×C₆₇	—	2	8	no	—
223	C₂×C₄×C₆₇	—	2	4	no	—
224	C₂×C₄×C₆₇	—	2	8	no	—
225	C₂×C₄×C₆₇	—	2	8	no	—
226	C₂×C₄×C₆₇	—	2	4	no	—
227	C₂×C₄×C₆₇	—	2	4	no	—
228	C₂×C₄×C₆₇	—	2	4	no	—
229	C₂×C₄×C₆₇	—	4	4	no	—
230	C₂×C₄×C₆₇	—	4	4	no	—
231	C₂×C₄×C₆₇	—	4	8	no	—
232	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
233	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
234	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
235	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
236	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
237	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
238	C₂×C₄×C₆₇	2₋¹⁺⁴	1	4	yes	—
239	C₂²×C₆₇	—	2	2	no	—
240	C₂²×C₆₇	—	2	2	no	—
241	C₂²×C₆₇	—	2	2	no	—
242	C₂²×C₆₇	—	4	4	no	—
243	C₂²×C₆₇	—	4	2	no	—
244	C₂²×C₆₇	—	4	2	no	—
245	C₂²×C₆₇	—	4	2	no	—
246	C₂²×C₆₇	—	4	8	no	—
247	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
248	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
249	C₂²×C₆₇	#585	1	2	yes	—
250	C₂²×C₆₇	#585	1	2	yes	—
251	C₄×C₆₇	—	2	2	no	—
252	C₄×C₆₇	—	2	2	no	—
253	C₄×C₆₇	—	2	2	no	—
254	C₄×C₆₇	—	2	2	no	—
255	C₄×C₆₇	—	2	2	no	—
256	C₄×C₆₇	—	4	2	no	—
257	C₄×C₆₇	—	4	2	no	—
258	C₄×C₆₇	—	4	2	no	—
259	C₄×C₆₇	—	4	4	no	—
260	C₄×C₆₇	—	4	4	no	—
261	C₄×C₆₇	—	4	8	no	—
262	C₄×C₆₇	#536	1	4	yes	—
263	#36699	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
264	C₂×C₆₇	—	4	2	no	—
265	C₂×C₆₇	—	8	2	no	—
266	C₂×C₆₇	C₂×#536	1	1	yes	characteristic
267	C₂×C₆₇	#3449	1	1	yes	characteristic
268	C₂×C₆₇	#3449	1	1	yes	characteristic
269	C₂×C₆₇	#3531	1	4	yes	—
270	C₂×#533	C₂×C₆₇	1	1	yes	characteristic, complemented
271	C₂×#529	C₂×C₆₇	1	1	yes	characteristic
272	#2822	C₂×C₆₇	1	2	yes	complemented
273	#2821	C₂×C₆₇	1	2	yes	complemented
274	#2821	C₂×C₆₇	1	2	yes	complemented
275	#2820	C₂×C₆₇	1	2	yes	complemented
276	#2817	C₂×C₆₇	1	2	yes	complemented
277	#2816	C₂×C₆₇	1	2	yes	—
278	#2814	C₂×C₆₇	1	4	yes	complemented
279	#2399	C₂×C₆₇	1	2	yes	complemented
280	#2398	C₂×C₆₇	1	2	yes	complemented
281	#2396	C₂×C₆₇	1	4	yes	complemented
282	#2395	C₂×C₆₇	1	4	yes	complemented
283	#2283	C₂×C₆₇	1	1	yes	characteristic, complemented
284	C₆₇	#36699	1	1	yes	characteristic, direct product factor, Sylow
285	#533	—	2	8	no	—
286	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
287	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
288	#529	—	2	8	no	—
289	C₂×#126	C₂²×C₆₇	1	2	yes	—
290	C₂×#125	C₂²×C₆₇	1	2	yes	—
291	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
292	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
293	C₂×#123	C₂²×C₆₇	1	2	yes	—
294	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
295	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
296	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
297	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
298	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
299	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
300	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
301	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
302	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
303	C₂×#117	C₂²×C₆₇	1	2	yes	—
304	C₂×#117	C₂²×C₆₇	1	2	yes	—
305	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
306	#400	C₂²×C₆₇	1	4	yes	—
307	#400	C₂²×C₆₇	1	4	yes	—
308	#400	C₂²×C₆₇	1	4	yes	—
309	#399	C₂²×C₆₇	1	4	yes	—
310	#399	C₂²×C₆₇	1	4	yes	—
311	#398	C₂²×C₆₇	1	4	yes	—
312	#398	C₂²×C₆₇	1	4	yes	—
313	#398	C₂²×C₆₇	1	4	yes	—
314	#398	C₂²×C₆₇	1	8	yes	—
315	#397	C₂²×C₆₇	1	8	yes	—
316	#397	C₂²×C₆₇	1	4	yes	—
317	#397	C₂²×C₆₇	1	4	yes	—
318	#396	C₂²×C₆₇	1	8	yes	—
319	#395	C₂²×C₆₇	1	4	yes	—
320	#395	C₂²×C₆₇	1	4	yes	—
321	#395	C₂²×C₆₇	1	4	yes	—
322	#394	C₂²×C₆₇	1	4	yes	—
323	#394	C₂²×C₆₇	1	8	yes	—
324	#393	C₂²×C₆₇	1	4	yes	—
325	#393	C₂²×C₆₇	1	4	yes	—
326	#383	C₂²×C₆₇	1	2	yes	—
327	#382	C₂²×C₆₇	1	2	yes	—
328	#381	C₂²×C₆₇	1	2	yes	—
329	#380	C₂²×C₆₇	1	2	yes	—
330	C₄×C₄⋊C₄	C₂²×C₆₇	1	8	yes	—
331	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
332	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
333	C₄×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	—
334	#376	C₂²×C₆₇	1	2	yes	—
335	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
336	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
337	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
338	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
339	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
340	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
341	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
342	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
343	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
344	C₄⋊Q₈	—	2	4	no	—
345	C₄⋊Q₈	—	2	4	no	—
346	#126	—	2	8	no	—
347	#125	—	2	4	no	—
348	#125	—	2	8	no	—
349	#124	—	2	4	no	—
350	#123	—	2	8	no	—
351	C₂²⋊Q₈	—	2	4	no	—
352	C₂²⋊Q₈	—	2	8	no	—
353	C₂²⋊Q₈	—	2	8	no	—
354	#121	—	2	4	no	—
355	C₄×Q₈	—	2	4	no	—
356	C₄×Q₈	—	2	8	no	—
357	C₄×Q₈	—	2	8	no	—
358	C₄×D₄	—	2	8	no	—
359	C₄×D₄	—	2	4	no	—
360	#117	—	2	8	no	—
361	#117	—	2	8	no	—
362	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
363	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
364	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
365	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
366	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
367	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
368	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
369	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
370	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
371	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
372	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
373	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
374	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
375	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
376	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
377	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
378	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
379	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
380	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
381	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
382	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
383	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
384	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
385	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
386	C₂×C₄²	D₄×C₆₇	1	8	yes	—
387	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
388	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
389	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
390	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
391	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
392	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
393	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
394	#95	C₂³×C₆₇	1	4	yes	—
395	#95	C₂³×C₆₇	1	4	yes	—
396	#95	C₂³×C₆₇	1	4	yes	—
397	#95	C₂³×C₆₇	1	8	yes	—
398	#95	C₂³×C₆₇	1	8	yes	—
399	#95	C₂³×C₆₇	1	4	yes	—
400	#95	C₂³×C₆₇	1	8	yes	—
401	#95	C₂³×C₆₇	1	8	yes	—
402	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
403	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
404	C₂×Q₈	—	2	2	no	—
405	C₂×Q₈	—	2	4	no	—
406	C₂×Q₈	—	2	4	no	—
407	C₂×Q₈	—	4	2	no	—
408	C₂×Q₈	—	4	4	no	—
409	C₂×Q₈	—	4	4	no	—
410	C₂×D₄	—	2	4	no	—
411	C₂×D₄	—	2	2	no	—
412	C₂×D₄	—	4	2	no	—
413	C₂×D₄	—	4	4	no	—
414	C₂²×C₄	—	2	4	no	—
415	C₂²×C₄	—	2	4	no	—
416	C₂²×C₄	—	2	8	no	—
417	C₂²×C₄	—	4	2	no	—
418	C₂²×C₄	—	4	2	no	—
419	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
420	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
421	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
422	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
423	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
424	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
425	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
426	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
427	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
428	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
429	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
430	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
431	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
432	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
433	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
434	C₄⋊C₄	—	2	4	no	—
435	C₄⋊C₄	—	2	4	no	—
436	C₄⋊C₄	—	2	4	no	—
437	C₄⋊C₄	—	2	4	no	—
438	C₄⋊C₄	—	2	2	no	—
439	C₄⋊C₄	—	2	4	no	—
440	C₄⋊C₄	—	2	2	no	—
441	C₄⋊C₄	—	2	8	no	—
442	C₄⋊C₄	—	2	8	no	—
443	C₄⋊C₄	—	2	4	no	—
444	C₄⋊C₄	—	2	4	no	—
445	C₄⋊C₄	—	2	4	no	—
446	C₄⋊C₄	—	4	4	no	—
447	C₄⋊C₄	—	4	8	no	—
448	C₄⋊C₄	—	4	8	no	—
449	C₂²⋊C₄	—	2	8	no	—
450	C₂²⋊C₄	—	2	4	no	—
451	C₂²⋊C₄	—	2	4	no	—
452	C₂²⋊C₄	—	2	4	no	—
453	C₂²⋊C₄	—	2	4	no	—
454	C₂²⋊C₄	—	2	4	no	—
455	C₂²⋊C₄	—	4	4	no	—
456	C₂²⋊C₄	—	4	8	no	—
457	C₂²⋊C₄	—	4	8	no	—
458	C₄²	—	2	4	no	—
459	C₄²	—	2	8	no	—
460	C₄²	—	2	8	no	—
461	C₄²	C₂×D₄×C₆₇	1	4	yes	—
462	C₄²	C₂×D₄×C₆₇	1	4	yes	—
463	C₄²	C₂×D₄×C₆₇	1	4	yes	—
464	C₄²	C₂×D₄×C₆₇	1	4	yes	—
465	C₂³	—	2	2	no	—
466	C₂³	—	2	2	no	—
467	C₂³	—	2	2	no	—
468	C₂³	—	2	2	no	—
469	C₂³	—	2	4	no	—
470	C₂³	—	2	4	no	—
471	C₂³	—	2	4	no	—
472	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
473	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
474	Q₈	—	4	4	no	—
475	Q₈	—	4	8	no	—
476	Q₈	—	4	8	no	—
477	D₄	—	4	4	no	—
478	D₄	—	4	8	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	2	8	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	4	no	—
488	C₂×C₄	—	2	4	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	8	no	—
491	C₂×C₄	—	2	4	no	—
492	C₂×C₄	—	2	8	no	—
493	C₂×C₄	—	2	4	no	—
494	C₂×C₄	—	2	8	no	—
495	C₂×C₄	—	2	8	no	—
496	C₂×C₄	—	2	4	no	—
497	C₂×C₄	—	2	4	no	—
498	C₂×C₄	—	2	4	no	—
499	C₂×C₄	—	4	4	no	—
500	C₂×C₄	—	4	4	no	—
501	C₂×C₄	—	4	8	no	—
502	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
503	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
504	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
505	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
506	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
507	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
508	C₂×C₄	2₋¹⁺⁴×C₆₇	1	4	yes	—
509	C₂²	—	2	2	no	—
510	C₂²	—	2	2	no	—
511	C₂²	—	2	2	no	—
512	C₂²	—	4	4	no	—
513	C₂²	—	4	8	no	—
514	C₂²	—	4	2	no	—
515	C₂²	—	4	2	no	—
516	C₂²	—	4	2	no	—
517	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
518	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	2	yes	—
519	C₂²	#585×C₆₇	1	2	yes	—
520	C₂²	#585×C₆₇	1	2	yes	—
521	C₄	—	2	2	no	—
522	C₄	—	2	2	no	—
523	C₄	—	2	2	no	—
524	C₄	—	2	2	no	—
525	C₄	—	2	2	no	—
526	C₄	—	4	8	no	—
527	C₄	—	4	2	no	—
528	C₄	—	4	2	no	—
529	C₄	—	4	2	no	—
530	C₄	—	4	4	no	—
531	C₄	—	4	4	no	—
532	C₄	#536×C₆₇	1	4	yes	—
533	C₂	—	4	2	no	—
534	C₂	—	8	2	no	—
535	C₂	C₂×#536×C₆₇	1	1	yes	characteristic
536	C₂	C₆₇×#3449	1	1	yes	characteristic
537	C₂	C₆₇×#3449	1	1	yes	characteristic
538	C₂	C₆₇×#3531	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
socle	C₂³×C₆₇	C₂⁵
prime radical	C₂×#121×C₆₇	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#533×C₆₇	C₂	semidirect
3. C₆₇×#2822	C₂	semidirect
4. C₆₇×#2821	C₂	semidirect
5. C₆₇×#2821	C₂	semidirect
6. C₆₇×#2820	C₂	semidirect
7. C₆₇×#2817	C₂	semidirect
9. C₆₇×#2814	C₂	semidirect
10. C₆₇×#2399	C₂	semidirect
11. C₆₇×#2398	C₂	semidirect
12. C₆₇×#2396	C₂	semidirect
13. C₆₇×#2395	C₂	semidirect
14. C₆₇×#2283	C₂	semidirect
263. #36699	C₆₇	direct (N × H)
270. C₂×#533	C₂×C₆₇	semidirect
272. #2822	C₂×C₆₇	semidirect
273. #2821	C₂×C₆₇	semidirect
274. #2821	C₂×C₆₇	semidirect
275. #2820	C₂×C₆₇	semidirect
276. #2817	C₂×C₆₇	semidirect
278. #2814	C₂×C₆₇	semidirect
279. #2399	C₂×C₆₇	semidirect
280. #2398	C₂×C₆₇	semidirect
281. #2396	C₂×C₆₇	semidirect
282. #2395	C₂×C₆₇	semidirect
283. #2283	C₂×C₆₇	semidirect
284. C₆₇	#36699	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#533×C₆₇	2
C₂×#529×C₆₇	2
C₆₇×#2822	2
C₆₇×#2821	2
C₆₇×#2820	2
C₆₇×#2817	2
C₆₇×#2816	2
C₆₇×#2814	2
C₆₇×#2399	2
C₆₇×#2398	2
C₆₇×#2396	2
C₆₇×#2395	2
C₆₇×#2283	2
#36699	67

Subgroups — C67×#36699 #30715950

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#36699 #30715950