Subgroups — C₆₇×#59668 #30738919

Order 17152.

1668 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2806	C₂	1	4	yes	maximal
2	C₆₇×#2584	C₂	1	2	yes	maximal
3	C₆₇×#2552	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2551	C₂	1	2	yes	maximal
5	C₆₇×#2551	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#2550	C₂	1	2	yes	maximal
7	C₆₇×#2546	C₂	1	2	yes	maximal
8	C₆₇×#2507	C₂	1	2	yes	maximal
9	C₆₇×#2507	C₂	1	2	yes	maximal
10	C₆₇×#2451	C₂	1	4	yes	maximal
11	C₆₇×#2370	C₂	1	2	yes	maximal
12	C₆₇×#2369	C₂	1	2	yes	maximal
13	C₆₇×#2369	C₂	1	1	yes	maximal, characteristic
14	C₆₇×#2369	C₂	1	1	yes	maximal, characteristic
15	C₄×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
16	C₄×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
17	C₄×#125×C₆₇	C₂	1	1	yes	maximal, characteristic
18	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄⋊Q₈×C₆₇	C₂²	1	4	yes	—
22	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#125×C₆₇	C₂²	1	2	yes	—
24	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
27	#398×C₆₇	C₂²	1	4	yes	—
28	#398×C₆₇	C₂²	1	4	yes	—
29	#398×C₆₇	C₂²	1	4	yes	—
30	#398×C₆₇	C₂²	1	4	yes	—
31	#398×C₆₇	C₂²	1	4	yes	—
32	#395×C₆₇	C₂²	1	4	yes	—
33	#395×C₆₇	C₂²	1	4	yes	—
34	#395×C₆₇	C₂²	1	4	yes	—
35	#391×C₆₇	C₂²	1	4	yes	—
36	#391×C₆₇	C₂²	1	4	yes	—
37	#391×C₆₇	C₂²	1	2	yes	—
38	#391×C₆₇	C₂²	1	4	yes	—
39	#391×C₆₇	C₂²	1	4	yes	—
40	#391×C₆₇	C₂²	1	2	yes	—
41	#389×C₆₇	C₂²	1	2	yes	—
42	#389×C₆₇	C₂²	1	1	yes	characteristic
43	#389×C₆₇	C₂²	1	2	yes	—
44	#389×C₆₇	C₂²	1	4	yes	—
45	#389×C₆₇	C₂²	1	2	yes	—
46	#389×C₆₇	C₂²	1	2	yes	—
47	#389×C₆₇	C₂²	1	2	yes	—
48	#389×C₆₇	C₂²	1	4	yes	—
49	#389×C₆₇	C₂²	1	1	yes	characteristic
50	#389×C₆₇	C₂²	1	2	yes	—
51	#389×C₆₇	C₂²	1	2	yes	—
52	#389×C₆₇	C₂²	1	2	yes	—
53	#389×C₆₇	C₂²	1	2	yes	—
54	#389×C₆₇	C₂²	1	2	yes	—
55	#389×C₆₇	C₂²	1	4	yes	—
56	#389×C₆₇	C₂²	1	4	yes	—
57	#389×C₆₇	C₂²	1	2	yes	—
58	#389×C₆₇	C₂²	1	1	yes	characteristic
59	#389×C₆₇	C₂²	1	1	yes	characteristic
60	#387×C₆₇	C₂²	1	4	yes	—
61	#387×C₆₇	C₂²	1	4	yes	—
62	#387×C₆₇	C₂²	1	2	yes	—
63	#387×C₆₇	C₂²	1	2	yes	—
64	#384×C₆₇	C₂²	1	1	yes	characteristic
65	#384×C₆₇	C₂²	1	2	yes	—
66	#384×C₆₇	C₂²	1	1	yes	characteristic
67	#382×C₆₇	C₂²	1	1	yes	characteristic
68	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
69	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
70	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
71	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
72	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
73	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
74	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
75	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
76	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
77	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
78	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
79	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
80	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
81	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
82	C₄³×C₆₇	C₂²	1	1	yes	characteristic
83	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
84	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
85	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
86	C₄⋊Q₈×C₆₇	—	2	4	no	—
87	C₄⋊Q₈×C₆₇	—	2	8	no	—
88	C₄⋊Q₈×C₆₇	—	4	4	no	—
89	C₄⋊Q₈×C₆₇	—	4	8	no	—
90	C₄⋊Q₈×C₆₇	Q₈	1	8	yes	complemented
91	#125×C₆₇	—	4	4	no	—
92	#125×C₆₇	Q₈	1	8	yes	complemented
93	C₄×Q₈×C₆₇	—	4	4	no	—
94	C₄×Q₈×C₆₇	—	4	4	no	—
95	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	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	2	yes	—
100	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
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	4	yes	—
104	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
107	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
110	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
111	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
112	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
113	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
114	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
115	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
116	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
117	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
118	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
119	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
120	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
121	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
122	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
123	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
124	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
125	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
126	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
128	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
129	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
130	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
131	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
132	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
133	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
134	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
135	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
136	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
137	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
138	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
139	#95×C₆₇	C₂³	1	2	yes	—
140	#95×C₆₇	C₂³	1	2	yes	—
141	#95×C₆₇	C₂³	1	2	yes	—
142	#95×C₆₇	C₂³	1	2	yes	—
143	#95×C₆₇	C₂³	1	4	yes	—
144	#95×C₆₇	C₂³	1	2	yes	—
145	#95×C₆₇	C₂³	1	2	yes	—
146	#95×C₆₇	C₂³	1	2	yes	—
147	#95×C₆₇	C₂³	1	4	yes	—
148	#95×C₆₇	C₂³	1	4	yes	—
149	#95×C₆₇	C₂³	1	4	yes	—
150	#95×C₆₇	C₂³	1	2	yes	—
151	#95×C₆₇	C₂³	1	4	yes	—
152	#95×C₆₇	C₂³	1	4	yes	—
153	#95×C₆₇	C₂³	1	2	yes	—
154	#95×C₆₇	C₂³	1	2	yes	—
155	#95×C₆₇	C₂³	1	4	yes	—
156	#95×C₆₇	C₂³	1	4	yes	—
157	C₂×Q₈×C₆₇	—	2	4	no	—
158	C₂×Q₈×C₆₇	—	2	8	no	—
159	C₂×Q₈×C₆₇	—	2	4	no	—
160	C₂×Q₈×C₆₇	—	4	2	no	—
161	C₂×Q₈×C₆₇	—	4	2	no	—
162	C₂×Q₈×C₆₇	—	4	2	no	—
163	C₂×Q₈×C₆₇	—	4	4	no	—
164	C₂×Q₈×C₆₇	—	4	4	no	—
165	C₂×Q₈×C₆₇	C₂×Q₈	1	8	yes	—
166	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
167	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
168	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
169	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
170	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
171	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
172	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
173	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
174	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
176	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
177	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
178	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
179	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
180	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
181	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
182	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
183	C₄⋊C₄×C₆₇	—	2	4	no	—
184	C₄⋊C₄×C₆₇	—	2	8	no	—
185	C₄⋊C₄×C₆₇	—	2	2	no	—
186	C₄⋊C₄×C₆₇	—	2	4	no	—
187	C₄⋊C₄×C₆₇	—	2	8	no	—
188	C₄⋊C₄×C₆₇	—	2	2	no	—
189	C₄⋊C₄×C₆₇	—	2	4	no	—
190	C₄⋊C₄×C₆₇	—	2	4	no	—
191	C₄⋊C₄×C₆₇	—	2	2	no	—
192	C₄⋊C₄×C₆₇	—	2	2	no	—
193	C₄⋊C₄×C₆₇	—	2	2	no	—
194	C₄⋊C₄×C₆₇	—	4	2	no	—
195	C₄⋊C₄×C₆₇	—	4	2	no	—
196	C₄⋊C₄×C₆₇	—	4	4	no	—
197	C₄⋊C₄×C₆₇	—	4	2	no	—
198	C₄⋊C₄×C₆₇	—	4	2	no	—
199	C₄⋊C₄×C₆₇	—	4	2	no	—
200	C₄⋊C₄×C₆₇	—	4	2	no	—
201	C₄⋊C₄×C₆₇	—	4	4	no	—
202	C₄⋊C₄×C₆₇	—	4	4	no	—
203	C₄⋊C₄×C₆₇	—	4	4	no	—
204	C₄⋊C₄×C₆₇	—	4	4	no	—
205	C₄⋊C₄×C₆₇	—	4	4	no	—
206	C₄⋊C₄×C₆₇	C₂×Q₈	1	8	yes	—
207	C₄⋊C₄×C₆₇	C₂×Q₈	1	8	yes	—
208	C₄⋊C₄×C₆₇	C₂×Q₈	1	4	yes	—
209	C₄⋊C₄×C₆₇	C₂×Q₈	1	4	yes	—
210	C₄⋊C₄×C₆₇	C₂×Q₈	1	8	yes	—
211	C₄⋊C₄×C₆₇	C₂×Q₈	1	8	yes	—
212	C₄⋊C₄×C₆₇	#41	1	4	yes	—
213	C₄²×C₆₇	—	2	2	no	—
214	C₄²×C₆₇	—	2	4	no	—
215	C₄²×C₆₇	—	2	2	no	—
216	C₄²×C₆₇	—	2	2	no	—
217	C₄²×C₆₇	—	2	4	no	—
218	C₄²×C₆₇	—	2	4	no	—
219	C₄²×C₆₇	—	2	2	no	—
220	C₄²×C₆₇	—	2	4	no	—
221	C₄²×C₆₇	—	4	4	no	—
222	C₄²×C₆₇	—	4	1	no	—
223	C₄²×C₆₇	—	4	4	no	—
224	C₄²×C₆₇	—	4	4	no	—
225	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
226	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
227	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
228	Q₈×C₆₇	—	4	8	no	—
229	Q₈×C₆₇	—	4	8	no	—
230	Q₈×C₆₇	—	8	8	no	—
231	C₂×C₄×C₆₇	—	2	2	no	—
232	C₂×C₄×C₆₇	—	2	1	no	—
233	C₂×C₄×C₆₇	—	2	1	no	—
234	C₂×C₄×C₆₇	—	2	1	no	—
235	C₂×C₄×C₆₇	—	2	1	no	—
236	C₂×C₄×C₆₇	—	2	1	no	—
237	C₂×C₄×C₆₇	—	2	2	no	—
238	C₂×C₄×C₆₇	—	2	2	no	—
239	C₂×C₄×C₆₇	—	2	4	no	—
240	C₂×C₄×C₆₇	—	2	2	no	—
241	C₂×C₄×C₆₇	—	2	1	no	—
242	C₂×C₄×C₆₇	—	2	2	no	—
243	C₂×C₄×C₆₇	—	2	2	no	—
244	C₂×C₄×C₆₇	—	2	1	no	—
245	C₂×C₄×C₆₇	—	2	4	no	—
246	C₂×C₄×C₆₇	—	2	1	no	—
247	C₂×C₄×C₆₇	—	2	2	no	—
248	C₂×C₄×C₆₇	—	2	2	no	—
249	C₂×C₄×C₆₇	—	2	4	no	—
250	C₂×C₄×C₆₇	—	2	1	no	—
251	C₂×C₄×C₆₇	—	2	2	no	—
252	C₂×C₄×C₆₇	—	2	2	no	—
253	C₂×C₄×C₆₇	—	2	2	no	—
254	C₂×C₄×C₆₇	—	2	4	no	—
255	C₂×C₄×C₆₇	—	2	2	no	—
256	C₂×C₄×C₆₇	—	2	4	no	—
257	C₂×C₄×C₆₇	—	2	2	no	—
258	C₂×C₄×C₆₇	—	2	2	no	—
259	C₂×C₄×C₆₇	—	2	2	no	—
260	C₂×C₄×C₆₇	—	2	1	no	—
261	C₂×C₄×C₆₇	—	2	2	no	—
262	C₂×C₄×C₆₇	—	2	2	no	—
263	C₂×C₄×C₆₇	—	2	2	no	—
264	C₂×C₄×C₆₇	—	2	2	no	—
265	C₂×C₄×C₆₇	—	2	1	no	—
266	C₂×C₄×C₆₇	—	2	2	no	—
267	C₂×C₄×C₆₇	—	2	2	no	—
268	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
269	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
270	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
271	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
272	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
273	C₂×C₄×C₆₇	C₂²×Q₈	1	8	yes	—
274	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
275	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
276	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
277	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
278	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
279	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
280	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
281	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
282	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
283	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
284	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
285	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
286	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
287	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
288	C₂²×C₆₇	#585	1	1	yes	characteristic
289	C₄×C₆₇	—	2	4	no	—
290	C₄×C₆₇	—	2	2	no	—
291	C₄×C₆₇	—	2	8	no	—
292	C₄×C₆₇	—	2	2	no	—
293	C₄×C₆₇	—	2	2	no	—
294	C₄×C₆₇	—	2	4	no	—
295	C₄×C₆₇	—	2	4	no	—
296	C₄×C₆₇	—	2	2	no	—
297	C₄×C₆₇	—	4	2	no	—
298	C₄×C₆₇	—	4	4	no	—
299	C₄×C₆₇	—	4	2	no	—
300	C₄×C₆₇	—	4	1	no	—
301	C₄×C₆₇	—	4	2	no	—
302	C₄×C₆₇	—	4	2	no	—
303	C₄×C₆₇	—	4	2	no	—
304	C₄×C₆₇	#557	1	4	yes	—
305	C₄×C₆₇	Q₈²	1	4	yes	—
306	#59668	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
307	C₂×C₆₇	C₂×#557	1	1	yes	characteristic
308	C₂×C₆₇	C₂×Q₈²	1	1	yes	characteristic
309	C₂×C₆₇	Q₈×#41	1	1	yes	characteristic
310	C₂×C₆₇	#3501	1	1	yes	characteristic
311	C₂×C₆₇	#3503	1	1	yes	characteristic
312	C₂×C₆₇	#3519	1	1	yes	characteristic
313	C₂×C₆₇	#3547	1	1	yes	characteristic
314	#2806	C₂×C₆₇	1	4	yes	—
315	#2584	C₂×C₆₇	1	2	yes	—
316	#2552	C₂×C₆₇	1	1	yes	characteristic
317	#2551	C₂×C₆₇	1	2	yes	—
318	#2551	C₂×C₆₇	1	1	yes	characteristic
319	#2550	C₂×C₆₇	1	2	yes	—
320	#2546	C₂×C₆₇	1	2	yes	—
321	#2507	C₂×C₆₇	1	2	yes	—
322	#2507	C₂×C₆₇	1	2	yes	—
323	#2451	C₂×C₆₇	1	4	yes	—
324	#2370	C₂×C₆₇	1	2	yes	—
325	#2369	C₂×C₆₇	1	2	yes	—
326	#2369	C₂×C₆₇	1	1	yes	characteristic
327	#2369	C₂×C₆₇	1	1	yes	characteristic
328	C₄×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic
329	C₄×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic
330	C₄×#125	C₂×C₆₇	1	1	yes	characteristic
331	C₆₇	#59668	1	1	yes	characteristic, direct product factor, Sylow
332	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
333	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
334	C₂×C₄⋊Q₈	C₂²×C₆₇	1	4	yes	—
335	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
336	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
337	C₂×#125	C₂²×C₆₇	1	2	yes	—
338	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
339	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
340	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
341	#398	C₂²×C₆₇	1	4	yes	—
342	#398	C₂²×C₆₇	1	4	yes	—
343	#398	C₂²×C₆₇	1	4	yes	—
344	#398	C₂²×C₆₇	1	4	yes	—
345	#398	C₂²×C₆₇	1	4	yes	—
346	#395	C₂²×C₆₇	1	4	yes	—
347	#395	C₂²×C₆₇	1	4	yes	—
348	#395	C₂²×C₆₇	1	4	yes	—
349	#391	C₂²×C₆₇	1	2	yes	—
350	#391	C₂²×C₆₇	1	2	yes	—
351	#391	C₂²×C₆₇	1	4	yes	—
352	#391	C₂²×C₆₇	1	4	yes	—
353	#391	C₂²×C₆₇	1	4	yes	—
354	#391	C₂²×C₆₇	1	4	yes	—
355	#389	C₂²×C₆₇	1	1	yes	characteristic
356	#389	C₂²×C₆₇	1	1	yes	characteristic
357	#389	C₂²×C₆₇	1	2	yes	—
358	#389	C₂²×C₆₇	1	1	yes	characteristic
359	#389	C₂²×C₆₇	1	1	yes	characteristic
360	#389	C₂²×C₆₇	1	2	yes	—
361	#389	C₂²×C₆₇	1	2	yes	—
362	#389	C₂²×C₆₇	1	2	yes	—
363	#389	C₂²×C₆₇	1	4	yes	—
364	#389	C₂²×C₆₇	1	2	yes	—
365	#389	C₂²×C₆₇	1	4	yes	—
366	#389	C₂²×C₆₇	1	2	yes	—
367	#389	C₂²×C₆₇	1	2	yes	—
368	#389	C₂²×C₆₇	1	2	yes	—
369	#389	C₂²×C₆₇	1	2	yes	—
370	#389	C₂²×C₆₇	1	2	yes	—
371	#389	C₂²×C₆₇	1	4	yes	—
372	#389	C₂²×C₆₇	1	4	yes	—
373	#389	C₂²×C₆₇	1	2	yes	—
374	#387	C₂²×C₆₇	1	2	yes	—
375	#387	C₂²×C₆₇	1	2	yes	—
376	#387	C₂²×C₆₇	1	4	yes	—
377	#387	C₂²×C₆₇	1	4	yes	—
378	#384	C₂²×C₆₇	1	1	yes	characteristic
379	#384	C₂²×C₆₇	1	1	yes	characteristic
380	#384	C₂²×C₆₇	1	2	yes	—
381	#382	C₂²×C₆₇	1	1	yes	characteristic
382	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
383	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
384	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
385	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
386	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
387	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
388	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
389	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
390	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
391	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
392	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
393	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
394	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
395	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
396	C₄³	C₂²×C₆₇	1	1	yes	characteristic
397	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
398	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
399	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
400	C₄⋊Q₈	—	2	4	no	—
401	C₄⋊Q₈	—	2	8	no	—
402	C₄⋊Q₈	—	4	4	no	—
403	C₄⋊Q₈	—	4	8	no	—
404	C₄⋊Q₈	Q₈×C₆₇	1	8	yes	complemented
405	#125	—	4	4	no	—
406	#125	Q₈×C₆₇	1	8	yes	complemented
407	C₄×Q₈	—	4	4	no	—
408	C₄×Q₈	—	4	4	no	—
409	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
410	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
411	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
412	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
413	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
414	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
415	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
416	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
417	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
418	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
419	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
420	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
421	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
422	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
423	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
424	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
425	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
426	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
427	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
428	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
429	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
430	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
431	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
432	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
433	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
434	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
435	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
436	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
437	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
438	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
439	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
440	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
441	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
442	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
443	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
444	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
445	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
446	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
447	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
448	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
449	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
450	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
451	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
452	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
453	#95	C₂³×C₆₇	1	2	yes	—
454	#95	C₂³×C₆₇	1	2	yes	—
455	#95	C₂³×C₆₇	1	2	yes	—
456	#95	C₂³×C₆₇	1	2	yes	—
457	#95	C₂³×C₆₇	1	4	yes	—
458	#95	C₂³×C₆₇	1	2	yes	—
459	#95	C₂³×C₆₇	1	2	yes	—
460	#95	C₂³×C₆₇	1	2	yes	—
461	#95	C₂³×C₆₇	1	4	yes	—
462	#95	C₂³×C₆₇	1	4	yes	—
463	#95	C₂³×C₆₇	1	4	yes	—
464	#95	C₂³×C₆₇	1	2	yes	—
465	#95	C₂³×C₆₇	1	4	yes	—
466	#95	C₂³×C₆₇	1	4	yes	—
467	#95	C₂³×C₆₇	1	2	yes	—
468	#95	C₂³×C₆₇	1	4	yes	—
469	#95	C₂³×C₆₇	1	2	yes	—
470	#95	C₂³×C₆₇	1	4	yes	—
471	C₂×Q₈	—	2	4	no	—
472	C₂×Q₈	—	2	8	no	—
473	C₂×Q₈	—	2	4	no	—
474	C₂×Q₈	—	4	2	no	—
475	C₂×Q₈	—	4	2	no	—
476	C₂×Q₈	—	4	2	no	—
477	C₂×Q₈	—	4	4	no	—
478	C₂×Q₈	—	4	4	no	—
479	C₂×Q₈	C₂×Q₈×C₆₇	1	8	yes	—
480	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
481	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
482	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
483	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
484	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
485	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
486	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
487	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
488	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
489	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
490	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
491	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
492	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
493	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
494	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
495	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
496	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
497	C₄⋊C₄	—	2	4	no	—
498	C₄⋊C₄	—	2	8	no	—
499	C₄⋊C₄	—	2	2	no	—
500	C₄⋊C₄	—	2	4	no	—
501	C₄⋊C₄	—	2	8	no	—
502	C₄⋊C₄	—	2	2	no	—
503	C₄⋊C₄	—	2	4	no	—
504	C₄⋊C₄	—	2	4	no	—
505	C₄⋊C₄	—	2	2	no	—
506	C₄⋊C₄	—	2	2	no	—
507	C₄⋊C₄	—	2	2	no	—
508	C₄⋊C₄	—	4	2	no	—
509	C₄⋊C₄	—	4	2	no	—
510	C₄⋊C₄	—	4	2	no	—
511	C₄⋊C₄	—	4	2	no	—
512	C₄⋊C₄	—	4	4	no	—
513	C₄⋊C₄	—	4	4	no	—
514	C₄⋊C₄	—	4	4	no	—
515	C₄⋊C₄	—	4	4	no	—
516	C₄⋊C₄	—	4	4	no	—
517	C₄⋊C₄	—	4	2	no	—
518	C₄⋊C₄	—	4	2	no	—
519	C₄⋊C₄	—	4	4	no	—
520	C₄⋊C₄	C₂×Q₈×C₆₇	1	8	yes	—
521	C₄⋊C₄	C₂×Q₈×C₆₇	1	8	yes	—
522	C₄⋊C₄	C₂×Q₈×C₆₇	1	4	yes	—
523	C₄⋊C₄	C₂×Q₈×C₆₇	1	4	yes	—
524	C₄⋊C₄	C₂×Q₈×C₆₇	1	8	yes	—
525	C₄⋊C₄	C₂×Q₈×C₆₇	1	8	yes	—
526	C₄⋊C₄	#41×C₆₇	1	4	yes	—
527	C₄²	—	2	2	no	—
528	C₄²	—	2	4	no	—
529	C₄²	—	2	2	no	—
530	C₄²	—	2	2	no	—
531	C₄²	—	2	4	no	—
532	C₄²	—	2	4	no	—
533	C₄²	—	2	2	no	—
534	C₄²	—	2	4	no	—
535	C₄²	—	4	4	no	—
536	C₄²	—	4	4	no	—
537	C₄²	—	4	4	no	—
538	C₄²	—	4	1	no	—
539	C₄²	C₂×Q₈×C₆₇	1	4	yes	—
540	C₄²	C₂×Q₈×C₆₇	1	4	yes	—
541	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
542	Q₈	—	4	8	no	—
543	Q₈	—	4	8	no	—
544	Q₈	—	8	8	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	1	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	4	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	2	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	1	no	—
559	C₂×C₄	—	2	4	no	—
560	C₂×C₄	—	2	1	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	4	no	—
564	C₂×C₄	—	2	1	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	4	no	—
569	C₂×C₄	—	2	2	no	—
570	C₂×C₄	—	2	4	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	2	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	2	no	—
576	C₂×C₄	—	2	2	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	2	2	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
583	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
584	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
585	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
586	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
587	C₂×C₄	C₂²×Q₈×C₆₇	1	8	yes	—
588	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
589	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
590	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
591	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
592	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
593	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
594	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
595	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
596	C₂²	C₂³×Q₈×C₆₇	1	1	yes	characteristic
597	C₂²	C₂³×Q₈×C₆₇	1	1	yes	characteristic
598	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
599	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
600	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
601	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
602	C₂²	#585×C₆₇	1	1	yes	characteristic
603	C₄	—	2	4	no	—
604	C₄	—	2	2	no	—
605	C₄	—	2	2	no	—
606	C₄	—	2	8	no	—
607	C₄	—	2	2	no	—
608	C₄	—	2	4	no	—
609	C₄	—	2	4	no	—
610	C₄	—	2	2	no	—
611	C₄	—	4	1	no	—
612	C₄	—	4	2	no	—
613	C₄	—	4	2	no	—
614	C₄	—	4	2	no	—
615	C₄	—	4	2	no	—
616	C₄	—	4	2	no	—
617	C₄	—	4	4	no	—
618	C₄	#557×C₆₇	1	4	yes	—
619	C₄	Q₈²×C₆₇	1	4	yes	—
620	C₂	C₂×#557×C₆₇	1	1	yes	characteristic
621	C₂	C₂×Q₈²×C₆₇	1	1	yes	characteristic
622	C₂	Q₈×#41×C₆₇	1	1	yes	characteristic
623	C₂	C₆₇×#3501	1	1	yes	characteristic
624	C₂	C₆₇×#3503	1	1	yes	characteristic
625	C₂	C₆₇×#3519	1	1	yes	characteristic
626	C₂	C₆₇×#3547	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
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
90. C₄⋊Q₈×C₆₇	Q₈	semidirect
92. #125×C₆₇	Q₈	semidirect
306. #59668	C₆₇	direct (N × H)
331. C₆₇	#59668	direct (N × H)
404. C₄⋊Q₈	Q₈×C₆₇	semidirect
406. #125	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×#2806	2
C₆₇×#2584	2
C₆₇×#2552	2
C₆₇×#2551	2
C₆₇×#2550	2
C₆₇×#2546	2
C₆₇×#2507	2
C₆₇×#2451	2
C₆₇×#2370	2
C₆₇×#2369	2
C₄×C₄⋊Q₈×C₆₇	2
C₄×#125×C₆₇	2
#59668	67

Subgroups — C67×#59668 #30738919

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#59668 #30738919