Subgroups — C₆₇×#59923 #30739174

Order 17152.

1818 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2811	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#2806	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2734	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2586	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2586	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2583	C₂	1	2	yes	maximal
7	C₆₇×#2549	C₂	1	4	yes	maximal
8	C₆₇×#2451	C₂	1	4	yes	maximal, complemented
9	C₆₇×#2434	C₂	1	2	yes	maximal
10	C₆₇×#2369	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2368	C₂	1	4	yes	maximal
12	C₄×C₂²⋊Q₈×C₆₇	C₂	1	2	yes	maximal
13	C₂×#391×C₆₇	C₂	1	1	yes	maximal, characteristic
14	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
17	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
18	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
19	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
21	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
23	#398×C₆₇	C₂²	1	4	yes	—
24	#398×C₆₇	C₂²	1	4	yes	—
25	#398×C₆₇	C₂²	1	2	yes	—
26	#398×C₆₇	C₂²	1	2	yes	—
27	#395×C₆₇	C₂²	1	4	yes	—
28	#395×C₆₇	C₂²	1	4	yes	—
29	#395×C₆₇	C₂²	1	4	yes	—
30	#395×C₆₇	C₂²	1	2	yes	—
31	#395×C₆₇	C₂²	1	2	yes	—
32	#395×C₆₇	C₂²	1	4	yes	—
33	#391×C₆₇	C₂²	1	4	yes	—
34	#391×C₆₇	C₂²	1	4	yes	—
35	#391×C₆₇	C₂²	1	1	yes	characteristic
36	#391×C₆₇	C₂²	1	1	yes	characteristic
37	#391×C₆₇	C₂²	1	1	yes	characteristic
38	#391×C₆₇	C₂²	1	4	yes	—
39	#391×C₆₇	C₂²	1	1	yes	characteristic
40	#391×C₆₇	C₂²	1	2	yes	—
41	#391×C₆₇	C₂²	1	4	yes	—
42	#391×C₆₇	C₂²	1	4	yes	—
43	#391×C₆₇	C₂²	1	2	yes	—
44	#391×C₆₇	C₂²	1	2	yes	—
45	#389×C₆₇	C₂²	1	4	yes	—
46	#389×C₆₇	C₂²	1	4	yes	—
47	#389×C₆₇	C₂²	1	2	yes	—
48	#389×C₆₇	C₂²	1	4	yes	—
49	#389×C₆₇	C₂²	1	4	yes	—
50	#387×C₆₇	C₂²	1	4	yes	—
51	#387×C₆₇	C₂²	1	4	yes	—
52	#387×C₆₇	C₂²	1	4	yes	—
53	#387×C₆₇	C₂²	1	4	yes	—
54	#387×C₆₇	C₂²	1	4	yes	—
55	#385×C₆₇	C₂²	1	4	yes	—
56	#385×C₆₇	C₂²	1	4	yes	—
57	#385×C₆₇	C₂²	1	4	yes	—
58	#380×C₆₇	C₂²	1	2	yes	—
59	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
60	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
61	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
62	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
63	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
64	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
65	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
66	C₂×#95×C₆₇	C₂²	1	4	yes	—
67	C₂²×Q₈×C₆₇	—	2	4	no	—
68	C₂²×Q₈×C₆₇	—	4	2	no	—
69	C₂²×Q₈×C₆₇	—	4	2	no	—
70	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
71	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
72	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
73	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
74	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
75	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
76	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
77	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
78	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
79	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
80	C₄⋊Q₈×C₆₇	—	2	8	no	—
81	C₄⋊Q₈×C₆₇	—	4	4	no	—
82	C₂²⋊Q₈×C₆₇	—	4	4	no	—
83	C₂²⋊Q₈×C₆₇	—	4	4	no	—
84	C₂²⋊Q₈×C₆₇	Q₈	1	16	yes	complemented
85	C₄×Q₈×C₆₇	—	4	4	no	—
86	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
87	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
88	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
89	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
90	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
92	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
93	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
94	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
95	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
97	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
98	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
99	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
101	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
102	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
103	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
104	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
105	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
106	C₂×C₄²×C₆₇	—	4	2	no	—
107	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
108	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
109	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
110	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
111	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
112	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
113	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
114	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
115	#95×C₆₇	C₂³	1	4	yes	—
116	#95×C₆₇	C₂³	1	4	yes	—
117	#95×C₆₇	C₂³	1	4	yes	—
118	#95×C₆₇	C₂³	1	4	yes	—
119	#95×C₆₇	C₂³	1	2	yes	—
120	#95×C₆₇	C₂³	1	4	yes	—
121	#95×C₆₇	C₂³	1	4	yes	—
122	#95×C₆₇	C₂³	1	4	yes	—
123	#95×C₆₇	C₂³	1	2	yes	—
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	4	yes	—
128	#95×C₆₇	C₂³	1	4	yes	—
129	#95×C₆₇	C₂³	1	4	yes	—
130	#95×C₆₇	C₂³	1	4	yes	—
131	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
132	C₂×Q₈×C₆₇	—	2	4	no	—
133	C₂×Q₈×C₆₇	—	2	2	no	—
134	C₂×Q₈×C₆₇	—	2	4	no	—
135	C₂×Q₈×C₆₇	—	2	4	no	—
136	C₂×Q₈×C₆₇	—	4	2	no	—
137	C₂×Q₈×C₆₇	—	4	1	no	—
138	C₂×Q₈×C₆₇	—	4	1	no	—
139	C₂×Q₈×C₆₇	—	4	1	no	—
140	C₂×Q₈×C₆₇	—	4	2	no	—
141	C₂×Q₈×C₆₇	—	4	1	no	—
142	C₂×Q₈×C₆₇	—	4	2	no	—
143	C₂×Q₈×C₆₇	—	4	2	no	—
144	C₂×Q₈×C₆₇	—	4	2	no	—
145	C₂×Q₈×C₆₇	—	4	2	no	—
146	C₂×Q₈×C₆₇	—	4	4	no	—
147	C₂×Q₈×C₆₇	—	4	16	no	—
148	C₂×Q₈×C₆₇	C₂×Q₈	1	8	yes	complemented
149	C₂×Q₈×C₆₇	#41	1	4	yes	—
150	C₂²×C₄×C₆₇	—	2	2	no	—
151	C₂²×C₄×C₆₇	—	2	2	no	—
152	C₂²×C₄×C₆₇	—	2	2	no	—
153	C₂²×C₄×C₆₇	—	2	4	no	—
154	C₂²×C₄×C₆₇	—	2	4	no	—
155	C₂²×C₄×C₆₇	—	2	4	no	—
156	C₂²×C₄×C₆₇	—	4	2	no	—
157	C₂²×C₄×C₆₇	—	4	2	no	—
158	C₂²×C₄×C₆₇	—	4	2	no	—
159	C₂²×C₄×C₆₇	—	4	4	no	—
160	C₂²×C₄×C₆₇	C₂×Q₈	1	8	yes	—
161	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
162	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
163	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
164	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
165	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
166	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
167	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
168	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
169	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
170	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
171	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
172	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
173	C₄⋊C₄×C₆₇	—	2	8	no	—
174	C₄⋊C₄×C₆₇	—	2	8	no	—
175	C₄⋊C₄×C₆₇	—	2	8	no	—
176	C₄⋊C₄×C₆₇	—	4	1	no	—
177	C₄⋊C₄×C₆₇	—	4	2	no	—
178	C₄⋊C₄×C₆₇	—	4	1	no	—
179	C₄⋊C₄×C₆₇	—	4	4	no	—
180	C₄⋊C₄×C₆₇	—	4	4	no	—
181	C₄⋊C₄×C₆₇	—	4	2	no	—
182	C₄⋊C₄×C₆₇	—	4	2	no	—
183	C₄⋊C₄×C₆₇	—	4	2	no	—
184	C₄⋊C₄×C₆₇	—	4	4	no	—
185	C₄⋊C₄×C₆₇	—	4	2	no	—
186	C₄⋊C₄×C₆₇	—	4	4	no	—
187	C₄⋊C₄×C₆₇	—	4	4	no	—
188	C₄⋊C₄×C₆₇	C₂×Q₈	1	8	yes	complemented
189	C₄⋊C₄×C₆₇	C₂×Q₈	1	16	yes	complemented
190	C₂²⋊C₄×C₆₇	—	2	8	no	—
191	C₂²⋊C₄×C₆₇	C₂×Q₈	1	16	yes	—
192	C₄²×C₆₇	—	2	4	no	—
193	C₄²×C₆₇	—	2	4	no	—
194	C₄²×C₆₇	—	4	1	no	—
195	C₄²×C₆₇	—	4	1	no	—
196	C₄²×C₆₇	—	4	4	no	—
197	C₄²×C₆₇	—	4	4	no	—
198	C₄²×C₆₇	—	4	4	no	—
199	C₄²×C₆₇	—	4	2	no	—
200	C₂³×C₆₇	—	2	1	no	—
201	C₂³×C₆₇	—	2	1	no	—
202	C₂³×C₆₇	—	2	2	no	—
203	C₂³×C₆₇	C₂²×Q₈	1	4	yes	—
204	C₂³×C₆₇	H(5,𝔽₂)	1	2	yes	—
205	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
206	Q₈×C₆₇	—	4	4	no	—
207	Q₈×C₆₇	—	4	8	no	—
208	Q₈×C₆₇	—	4	8	no	—
209	Q₈×C₆₇	—	4	4	no	—
210	Q₈×C₆₇	—	8	4	no	—
211	Q₈×C₆₇	—	8	4	no	—
212	C₂×C₄×C₆₇	—	2	8	no	—
213	C₂×C₄×C₆₇	—	2	4	no	—
214	C₂×C₄×C₆₇	—	2	1	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	1	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	2	no	—
223	C₂×C₄×C₆₇	—	2	4	no	—
224	C₂×C₄×C₆₇	—	2	2	no	—
225	C₂×C₄×C₆₇	—	2	4	no	—
226	C₂×C₄×C₆₇	—	2	2	no	—
227	C₂×C₄×C₆₇	—	2	4	no	—
228	C₂×C₄×C₆₇	—	2	2	no	—
229	C₂×C₄×C₆₇	—	2	4	no	—
230	C₂×C₄×C₆₇	—	2	4	no	—
231	C₂×C₄×C₆₇	—	2	4	no	—
232	C₂×C₄×C₆₇	—	2	2	no	—
233	C₂×C₄×C₆₇	—	2	2	no	—
234	C₂×C₄×C₆₇	—	2	2	no	—
235	C₂×C₄×C₆₇	—	2	2	no	—
236	C₂×C₄×C₆₇	—	2	4	no	—
237	C₂×C₄×C₆₇	—	2	4	no	—
238	C₂×C₄×C₆₇	—	2	4	no	—
239	C₂×C₄×C₆₇	—	2	4	no	—
240	C₂×C₄×C₆₇	—	4	8	no	—
241	C₂×C₄×C₆₇	—	4	8	no	—
242	C₂×C₄×C₆₇	—	8	4	no	—
243	C₂×C₄×C₆₇	C₂²×Q₈	1	8	yes	—
244	C₂×C₄×C₆₇	C₂²×Q₈	1	8	yes	—
245	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
246	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
247	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
248	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
249	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
250	C₂²×C₆₇	—	2	4	no	—
251	C₂²×C₆₇	—	2	2	no	—
252	C₂²×C₆₇	—	2	4	no	—
253	C₂²×C₆₇	—	2	2	no	—
254	C₂²×C₆₇	Q₈²	1	4	yes	—
255	C₂²×C₆₇	C₂³×Q₈	1	2	yes	—
256	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
257	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
258	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
259	C₂²×C₆₇	#585	1	2	yes	—
260	C₄×C₆₇	—	2	8	no	—
261	C₄×C₆₇	—	2	2	no	—
262	C₄×C₆₇	—	2	2	no	—
263	C₄×C₆₇	—	2	4	no	—
264	C₄×C₆₇	—	2	4	no	—
265	C₄×C₆₇	—	2	4	no	—
266	C₄×C₆₇	—	2	8	no	—
267	C₄×C₆₇	—	4	2	no	—
268	C₄×C₆₇	—	4	4	no	—
269	C₄×C₆₇	—	4	2	no	—
270	C₄×C₆₇	—	4	2	no	—
271	C₄×C₆₇	—	4	4	no	—
272	#59923	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
273	C₂×C₆₇	—	2	4	no	—
274	C₂×C₆₇	C₂×Q₈²	1	1	yes	characteristic
275	C₂×C₆₇	Q₈×#41	1	2	yes	—
276	C₂×C₆₇	#3490	1	1	yes	characteristic
277	C₂×C₆₇	#3501	1	2	yes	—
278	C₂×C₆₇	#3545	1	1	yes	characteristic
279	#2811	C₂×C₆₇	1	1	yes	characteristic, complemented
280	#2806	C₂×C₆₇	1	2	yes	complemented
281	#2734	C₂×C₆₇	1	1	yes	characteristic, complemented
282	#2586	C₂×C₆₇	1	4	yes	complemented
283	#2586	C₂×C₆₇	1	2	yes	complemented
284	#2583	C₂×C₆₇	1	2	yes	—
285	#2549	C₂×C₆₇	1	4	yes	—
286	#2451	C₂×C₆₇	1	4	yes	complemented
287	#2434	C₂×C₆₇	1	2	yes	—
288	#2369	C₂×C₆₇	1	2	yes	complemented
289	#2368	C₂×C₆₇	1	4	yes	—
290	C₄×C₂²⋊Q₈	C₂×C₆₇	1	2	yes	—
291	C₂×#391	C₂×C₆₇	1	1	yes	characteristic
292	C₆₇	#59923	1	1	yes	characteristic, direct product factor, Sylow
293	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
294	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
295	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
296	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
297	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
298	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
299	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
300	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
301	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
302	#398	C₂²×C₆₇	1	2	yes	—
303	#398	C₂²×C₆₇	1	2	yes	—
304	#398	C₂²×C₆₇	1	4	yes	—
305	#398	C₂²×C₆₇	1	4	yes	—
306	#395	C₂²×C₆₇	1	2	yes	—
307	#395	C₂²×C₆₇	1	2	yes	—
308	#395	C₂²×C₆₇	1	4	yes	—
309	#395	C₂²×C₆₇	1	4	yes	—
310	#395	C₂²×C₆₇	1	4	yes	—
311	#395	C₂²×C₆₇	1	4	yes	—
312	#391	C₂²×C₆₇	1	2	yes	—
313	#391	C₂²×C₆₇	1	4	yes	—
314	#391	C₂²×C₆₇	1	1	yes	characteristic
315	#391	C₂²×C₆₇	1	4	yes	—
316	#391	C₂²×C₆₇	1	4	yes	—
317	#391	C₂²×C₆₇	1	4	yes	—
318	#391	C₂²×C₆₇	1	1	yes	characteristic
319	#391	C₂²×C₆₇	1	2	yes	—
320	#391	C₂²×C₆₇	1	4	yes	—
321	#391	C₂²×C₆₇	1	1	yes	characteristic
322	#391	C₂²×C₆₇	1	1	yes	characteristic
323	#391	C₂²×C₆₇	1	2	yes	—
324	#389	C₂²×C₆₇	1	4	yes	—
325	#389	C₂²×C₆₇	1	4	yes	—
326	#389	C₂²×C₆₇	1	4	yes	—
327	#389	C₂²×C₆₇	1	4	yes	—
328	#389	C₂²×C₆₇	1	2	yes	—
329	#387	C₂²×C₆₇	1	4	yes	—
330	#387	C₂²×C₆₇	1	4	yes	—
331	#387	C₂²×C₆₇	1	4	yes	—
332	#387	C₂²×C₆₇	1	4	yes	—
333	#387	C₂²×C₆₇	1	4	yes	—
334	#385	C₂²×C₆₇	1	4	yes	—
335	#385	C₂²×C₆₇	1	4	yes	—
336	#385	C₂²×C₆₇	1	4	yes	—
337	#380	C₂²×C₆₇	1	2	yes	—
338	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
339	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
340	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
341	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
342	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
343	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
344	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
345	C₂×#95	C₂²×C₆₇	1	4	yes	—
346	C₂²×Q₈	—	2	4	no	—
347	C₂²×Q₈	—	4	2	no	—
348	C₂²×Q₈	—	4	2	no	—
349	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
350	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
351	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
352	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
353	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
354	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
355	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
356	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
357	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
358	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
359	C₄⋊Q₈	—	2	8	no	—
360	C₄⋊Q₈	—	4	4	no	—
361	C₂²⋊Q₈	—	4	4	no	—
362	C₂²⋊Q₈	—	4	4	no	—
363	C₂²⋊Q₈	Q₈×C₆₇	1	16	yes	complemented
364	C₄×Q₈	—	4	4	no	—
365	C₂×C₄⋊C₄	—	4	2	no	—
366	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
367	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
368	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
369	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
370	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
371	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
372	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
373	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
374	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
375	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
376	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
377	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
378	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
379	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
380	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
381	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
382	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
383	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
384	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
385	C₂×C₄²	—	4	2	no	—
386	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
387	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
388	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
389	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
390	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
391	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
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	4	yes	—
398	#95	C₂³×C₆₇	1	2	yes	—
399	#95	C₂³×C₆₇	1	4	yes	—
400	#95	C₂³×C₆₇	1	4	yes	—
401	#95	C₂³×C₆₇	1	2	yes	—
402	#95	C₂³×C₆₇	1	4	yes	—
403	#95	C₂³×C₆₇	1	4	yes	—
404	#95	C₂³×C₆₇	1	4	yes	—
405	#95	C₂³×C₆₇	1	4	yes	—
406	#95	C₂³×C₆₇	1	4	yes	—
407	#95	C₂³×C₆₇	1	4	yes	—
408	#95	C₂³×C₆₇	1	4	yes	—
409	#95	C₂³×C₆₇	1	4	yes	—
410	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
411	C₂×Q₈	—	2	4	no	—
412	C₂×Q₈	—	2	2	no	—
413	C₂×Q₈	—	2	4	no	—
414	C₂×Q₈	—	2	4	no	—
415	C₂×Q₈	—	4	2	no	—
416	C₂×Q₈	—	4	1	no	—
417	C₂×Q₈	—	4	1	no	—
418	C₂×Q₈	—	4	1	no	—
419	C₂×Q₈	—	4	2	no	—
420	C₂×Q₈	—	4	1	no	—
421	C₂×Q₈	—	4	2	no	—
422	C₂×Q₈	—	4	2	no	—
423	C₂×Q₈	—	4	2	no	—
424	C₂×Q₈	—	4	2	no	—
425	C₂×Q₈	—	4	4	no	—
426	C₂×Q₈	—	4	16	no	—
427	C₂×Q₈	C₂×Q₈×C₆₇	1	8	yes	complemented
428	C₂×Q₈	#41×C₆₇	1	4	yes	—
429	C₂²×C₄	—	2	2	no	—
430	C₂²×C₄	—	2	2	no	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	2	4	no	—
433	C₂²×C₄	—	2	4	no	—
434	C₂²×C₄	—	2	4	no	—
435	C₂²×C₄	—	4	2	no	—
436	C₂²×C₄	—	4	2	no	—
437	C₂²×C₄	—	4	2	no	—
438	C₂²×C₄	—	4	4	no	—
439	C₂²×C₄	C₂×Q₈×C₆₇	1	8	yes	—
440	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
441	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
442	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
443	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
444	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
445	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
446	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
447	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
448	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
449	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
450	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
451	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
452	C₄⋊C₄	—	2	8	no	—
453	C₄⋊C₄	—	2	8	no	—
454	C₄⋊C₄	—	2	8	no	—
455	C₄⋊C₄	—	4	1	no	—
456	C₄⋊C₄	—	4	2	no	—
457	C₄⋊C₄	—	4	1	no	—
458	C₄⋊C₄	—	4	2	no	—
459	C₄⋊C₄	—	4	2	no	—
460	C₄⋊C₄	—	4	4	no	—
461	C₄⋊C₄	—	4	2	no	—
462	C₄⋊C₄	—	4	4	no	—
463	C₄⋊C₄	—	4	4	no	—
464	C₄⋊C₄	—	4	4	no	—
465	C₄⋊C₄	—	4	4	no	—
466	C₄⋊C₄	—	4	2	no	—
467	C₄⋊C₄	C₂×Q₈×C₆₇	1	8	yes	complemented
468	C₄⋊C₄	C₂×Q₈×C₆₇	1	16	yes	complemented
469	C₂²⋊C₄	—	2	8	no	—
470	C₂²⋊C₄	C₂×Q₈×C₆₇	1	16	yes	—
471	C₄²	—	2	4	no	—
472	C₄²	—	2	4	no	—
473	C₄²	—	4	1	no	—
474	C₄²	—	4	1	no	—
475	C₄²	—	4	4	no	—
476	C₄²	—	4	2	no	—
477	C₄²	—	4	4	no	—
478	C₄²	—	4	4	no	—
479	C₂³	—	2	1	no	—
480	C₂³	—	2	1	no	—
481	C₂³	—	2	2	no	—
482	C₂³	C₂²×Q₈×C₆₇	1	4	yes	—
483	C₂³	H(5,𝔽₂)×C₆₇	1	2	yes	—
484	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
485	Q₈	—	4	4	no	—
486	Q₈	—	4	8	no	—
487	Q₈	—	4	8	no	—
488	Q₈	—	4	4	no	—
489	Q₈	—	8	4	no	—
490	Q₈	—	8	4	no	—
491	C₂×C₄	—	2	8	no	—
492	C₂×C₄	—	2	4	no	—
493	C₂×C₄	—	2	1	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	4	no	—
497	C₂×C₄	—	2	1	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	1	no	—
500	C₂×C₄	—	2	1	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	2	4	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	4	no	—
505	C₂×C₄	—	2	2	no	—
506	C₂×C₄	—	2	4	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	4	no	—
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	2	4	no	—
516	C₂×C₄	—	2	4	no	—
517	C₂×C₄	—	2	4	no	—
518	C₂×C₄	—	2	4	no	—
519	C₂×C₄	—	4	8	no	—
520	C₂×C₄	—	4	8	no	—
521	C₂×C₄	—	8	4	no	—
522	C₂×C₄	C₂²×Q₈×C₆₇	1	8	yes	—
523	C₂×C₄	C₂²×Q₈×C₆₇	1	8	yes	—
524	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
525	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
526	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
527	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
528	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
529	C₂²	—	2	2	no	—
530	C₂²	—	2	4	no	—
531	C₂²	—	2	2	no	—
532	C₂²	—	2	4	no	—
533	C₂²	Q₈²×C₆₇	1	4	yes	—
534	C₂²	C₂³×Q₈×C₆₇	1	2	yes	—
535	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
536	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
537	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
538	C₂²	#585×C₆₇	1	2	yes	—
539	C₄	—	2	8	no	—
540	C₄	—	2	2	no	—
541	C₄	—	2	2	no	—
542	C₄	—	2	4	no	—
543	C₄	—	2	4	no	—
544	C₄	—	2	4	no	—
545	C₄	—	2	8	no	—
546	C₄	—	4	2	no	—
547	C₄	—	4	2	no	—
548	C₄	—	4	4	no	—
549	C₄	—	4	2	no	—
550	C₄	—	4	4	no	—
551	C₂	—	2	4	no	—
552	C₂	C₂×Q₈²×C₆₇	1	1	yes	characteristic
553	C₂	Q₈×#41×C₆₇	1	2	yes	—
554	C₂	C₆₇×#3490	1	1	yes	characteristic
555	C₂	C₆₇×#3501	1	2	yes	—
556	C₂	C₆₇×#3545	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₆₇×#2811	C₂	semidirect
2. C₆₇×#2806	C₂	semidirect
3. C₆₇×#2734	C₂	semidirect
4. C₆₇×#2586	C₂	semidirect
5. C₆₇×#2586	C₂	semidirect
8. C₆₇×#2451	C₂	semidirect
10. C₆₇×#2369	C₂	semidirect
84. C₂²⋊Q₈×C₆₇	Q₈	semidirect
148. C₂×Q₈×C₆₇	C₂×Q₈	semidirect
188. C₄⋊C₄×C₆₇	C₂×Q₈	semidirect
189. C₄⋊C₄×C₆₇	C₂×Q₈	semidirect
272. #59923	C₆₇	direct (N × H)
279. #2811	C₂×C₆₇	semidirect
280. #2806	C₂×C₆₇	semidirect
281. #2734	C₂×C₆₇	semidirect
282. #2586	C₂×C₆₇	semidirect
283. #2586	C₂×C₆₇	semidirect
286. #2451	C₂×C₆₇	semidirect
288. #2369	C₂×C₆₇	semidirect
292. C₆₇	#59923	direct (N × H)
363. C₂²⋊Q₈	Q₈×C₆₇	semidirect
427. C₂×Q₈	C₂×Q₈×C₆₇	semidirect
467. C₄⋊C₄	C₂×Q₈×C₆₇	semidirect
468. C₄⋊C₄	C₂×Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2811	2
C₆₇×#2806	2
C₆₇×#2734	2
C₆₇×#2586	2
C₆₇×#2583	2
C₆₇×#2549	2
C₆₇×#2451	2
C₆₇×#2434	2
C₆₇×#2369	2
C₆₇×#2368	2
C₄×C₂²⋊Q₈×C₆₇	2
C₂×#391×C₆₇	2
#59923	67

Subgroups — C67×#59923 #30739174

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#59923 #30739174