Subgroups — C₆₇×#59828 #30739079

Order 17152.

2004 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#557×C₆₇	C₂	1	2	yes	maximal
2	C₆₇×#2726	C₂	1	4	yes	maximal
3	C₆₇×#2582	C₂	1	2	yes	maximal
4	C₆₇×#2552	C₂	1	2	yes	maximal
5	C₆₇×#2546	C₂	1	2	yes	maximal
6	C₆₇×#2506	C₂	1	4	yes	maximal
7	C₆₇×#2453	C₂	1	4	yes	maximal
8	C₆₇×#2446	C₂	1	4	yes	maximal
9	C₆₇×#2365	C₂	1	2	yes	maximal
10	Q₈×C₄⋊C₄×C₆₇	C₂	1	2	yes	maximal
11	C₄×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₄×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
13	C₄×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
14	#557×C₆₇	—	2	16	no	—
15	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Q₈×C₆₇	C₂²	1	4	yes	—
17	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#125×C₆₇	C₂²	1	8	yes	—
20	C₂×#125×C₆₇	C₂²	1	4	yes	—
21	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
23	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×Q₈×C₆₇	C₂²	1	4	yes	—
25	C₂×C₄×Q₈×C₆₇	C₂²	1	4	yes	—
26	#400×C₆₇	C₂²	1	8	yes	—
27	#398×C₆₇	C₂²	1	8	yes	—
28	#398×C₆₇	C₂²	1	8	yes	—
29	#398×C₆₇	C₂²	1	8	yes	—
30	#391×C₆₇	C₂²	1	4	yes	—
31	#391×C₆₇	C₂²	1	2	yes	—
32	#391×C₆₇	C₂²	1	2	yes	—
33	#391×C₆₇	C₂²	1	2	yes	—
34	#391×C₆₇	C₂²	1	4	yes	—
35	#389×C₆₇	C₂²	1	2	yes	—
36	#389×C₆₇	C₂²	1	2	yes	—
37	#389×C₆₇	C₂²	1	4	yes	—
38	#389×C₆₇	C₂²	1	4	yes	—
39	#389×C₆₇	C₂²	1	4	yes	—
40	#389×C₆₇	C₂²	1	2	yes	—
41	#389×C₆₇	C₂²	1	2	yes	—
42	#389×C₆₇	C₂²	1	2	yes	—
43	#389×C₆₇	C₂²	1	2	yes	—
44	#389×C₆₇	C₂²	1	4	yes	—
45	#389×C₆₇	C₂²	1	8	yes	—
46	#387×C₆₇	C₂²	1	8	yes	—
47	#387×C₆₇	C₂²	1	8	yes	—
48	#384×C₆₇	C₂²	1	4	yes	—
49	#384×C₆₇	C₂²	1	1	yes	characteristic
50	#384×C₆₇	C₂²	1	1	yes	characteristic
51	#384×C₆₇	C₂²	1	1	yes	characteristic
52	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
53	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
54	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
55	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
56	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
57	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
58	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
59	C₄³×C₆₇	C₂²	1	1	yes	characteristic
60	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
61	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
62	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
63	C₄⋊Q₈×C₆₇	—	2	16	no	—
64	C₄⋊Q₈×C₆₇	—	2	4	no	—
65	C₄⋊Q₈×C₆₇	D₄	1	8	yes	—
66	C₄⋊Q₈×C₆₇	Q₈	1	8	yes	complemented
67	#125×C₆₇	—	2	32	no	—
68	#125×C₆₇	—	2	16	no	—
69	C₄×Q₈×C₆₇	—	2	16	no	—
70	C₄×Q₈×C₆₇	—	2	16	no	—
71	C₄×Q₈×C₆₇	—	2	8	no	—
72	C₄×Q₈×C₆₇	—	2	8	no	—
73	C₄×Q₈×C₆₇	—	2	8	no	—
74	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
75	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
76	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
77	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
78	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
79	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
80	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
81	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
82	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
83	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
84	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
85	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
86	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
87	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
88	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
89	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
90	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
91	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
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×C₄²×C₆₇	C₂³	1	4	yes	—
97	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
98	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
99	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
100	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
106	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
107	#95×C₆₇	C₂³	1	4	yes	—
108	#95×C₆₇	C₂³	1	4	yes	—
109	#95×C₆₇	C₂³	1	4	yes	—
110	#95×C₆₇	C₂³	1	4	yes	—
111	#95×C₆₇	C₂³	1	4	yes	—
112	#95×C₆₇	C₂³	1	8	yes	—
113	#95×C₆₇	C₂³	1	4	yes	—
114	#95×C₆₇	C₂³	1	8	yes	—
115	#95×C₆₇	C₂³	1	8	yes	—
116	C₂×Q₈×C₆₇	—	2	4	no	—
117	C₂×Q₈×C₆₇	—	2	4	no	—
118	C₂×Q₈×C₆₇	—	2	4	no	—
119	C₂×Q₈×C₆₇	—	2	4	no	—
120	C₂×Q₈×C₆₇	—	2	4	no	—
121	C₂×Q₈×C₆₇	—	2	4	no	—
122	C₂×Q₈×C₆₇	—	2	4	no	—
123	C₂×Q₈×C₆₇	C₂×D₄	1	8	yes	—
124	C₂×Q₈×C₆₇	C₂×Q₈	1	8	yes	—
125	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
126	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
127	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
128	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
129	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
130	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
131	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
132	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
133	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
134	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
135	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
136	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
137	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
138	C₄⋊C₄×C₆₇	—	2	4	no	—
139	C₄⋊C₄×C₆₇	—	2	8	no	—
140	C₄⋊C₄×C₆₇	—	2	16	no	—
141	C₄⋊C₄×C₆₇	—	2	8	no	—
142	C₄⋊C₄×C₆₇	—	2	4	no	—
143	C₄⋊C₄×C₆₇	—	2	16	no	—
144	C₄⋊C₄×C₆₇	—	2	16	no	—
145	C₄⋊C₄×C₆₇	—	2	8	no	—
146	C₄⋊C₄×C₆₇	—	2	4	no	—
147	C₄⋊C₄×C₆₇	—	2	4	no	—
148	C₄⋊C₄×C₆₇	—	2	4	no	—
149	C₄⋊C₄×C₆₇	—	2	4	no	—
150	C₄⋊C₄×C₆₇	—	2	4	no	—
151	C₄⋊C₄×C₆₇	—	2	4	no	—
152	C₄⋊C₄×C₆₇	—	4	2	no	—
153	C₄⋊C₄×C₆₇	—	4	2	no	—
154	C₄⋊C₄×C₆₇	—	4	2	no	—
155	C₄⋊C₄×C₆₇	—	4	8	no	—
156	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
157	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
158	C₄⋊C₄×C₆₇	C₂×Q₈	1	8	yes	—
159	C₄⋊C₄×C₆₇	C₂×Q₈	1	8	yes	—
160	C₄²×C₆₇	—	2	16	no	—
161	C₄²×C₆₇	—	2	8	no	—
162	C₄²×C₆₇	—	2	8	no	—
163	C₄²×C₆₇	—	2	8	no	—
164	C₄²×C₆₇	—	2	2	no	—
165	C₄²×C₆₇	—	2	2	no	—
166	C₄²×C₆₇	—	2	2	no	—
167	C₄²×C₆₇	—	4	1	no	—
168	C₄²×C₆₇	C₂×D₄	1	4	yes	—
169	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
170	C₄²×C₆₇	#41	1	4	yes	—
171	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
172	Q₈×C₆₇	—	2	16	no	—
173	Q₈×C₆₇	—	2	16	no	—
174	Q₈×C₆₇	—	4	8	no	—
175	C₂×C₄×C₆₇	—	2	4	no	—
176	C₂×C₄×C₆₇	—	2	4	no	—
177	C₂×C₄×C₆₇	—	2	1	no	—
178	C₂×C₄×C₆₇	—	2	1	no	—
179	C₂×C₄×C₆₇	—	2	1	no	—
180	C₂×C₄×C₆₇	—	2	4	no	—
181	C₂×C₄×C₆₇	—	2	4	no	—
182	C₂×C₄×C₆₇	—	2	1	no	—
183	C₂×C₄×C₆₇	—	2	1	no	—
184	C₂×C₄×C₆₇	—	2	1	no	—
185	C₂×C₄×C₆₇	—	2	4	no	—
186	C₂×C₄×C₆₇	—	2	4	no	—
187	C₂×C₄×C₆₇	—	2	4	no	—
188	C₂×C₄×C₆₇	—	2	4	no	—
189	C₂×C₄×C₆₇	—	2	4	no	—
190	C₂×C₄×C₆₇	—	2	1	no	—
191	C₂×C₄×C₆₇	—	2	1	no	—
192	C₂×C₄×C₆₇	—	2	1	no	—
193	C₂×C₄×C₆₇	—	2	4	no	—
194	C₂×C₄×C₆₇	—	2	4	no	—
195	C₂×C₄×C₆₇	—	2	4	no	—
196	C₂×C₄×C₆₇	—	2	4	no	—
197	C₂×C₄×C₆₇	—	2	4	no	—
198	C₂×C₄×C₆₇	—	2	4	no	—
199	C₂×C₄×C₆₇	—	2	4	no	—
200	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
201	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
202	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
203	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
204	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
205	C₂×C₄×C₆₇	C₂²×Q₈	1	8	yes	—
206	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
207	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
208	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
209	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
210	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
211	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
212	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
213	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
214	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
215	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
216	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
217	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
218	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
219	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
220	C₂²×C₆₇	#585	1	1	yes	characteristic
221	C₄×C₆₇	—	2	8	no	—
222	C₄×C₆₇	—	2	8	no	—
223	C₄×C₆₇	—	2	2	no	—
224	C₄×C₆₇	—	2	2	no	—
225	C₄×C₆₇	—	2	2	no	—
226	C₄×C₆₇	—	4	4	no	—
227	C₄×C₆₇	—	4	4	no	—
228	C₄×C₆₇	—	4	4	no	—
229	C₄×C₆₇	—	4	4	no	—
230	C₄×C₆₇	—	4	1	no	—
231	C₄×C₆₇	#547	1	4	yes	—
232	C₄×C₆₇	D₄×Q₈	1	4	yes	—
233	C₄×C₆₇	#557	1	4	yes	complemented
234	#59828	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
235	C₂×C₆₇	C₂×#547	1	1	yes	characteristic
236	C₂×C₆₇	C₂×D₄×Q₈	1	1	yes	characteristic
237	C₂×C₆₇	C₂×#557	1	1	yes	characteristic
238	C₂×C₆₇	#3497	1	1	yes	characteristic
239	C₂×C₆₇	#3502	1	1	yes	characteristic
240	C₂×C₆₇	#3507	1	1	yes	characteristic
241	C₂×C₆₇	#3556	1	1	yes	characteristic
242	C₂×#557	C₂×C₆₇	1	2	yes	—
243	#2726	C₂×C₆₇	1	4	yes	—
244	#2582	C₂×C₆₇	1	2	yes	—
245	#2552	C₂×C₆₇	1	2	yes	—
246	#2546	C₂×C₆₇	1	2	yes	—
247	#2506	C₂×C₆₇	1	4	yes	—
248	#2453	C₂×C₆₇	1	4	yes	—
249	#2446	C₂×C₆₇	1	4	yes	—
250	#2365	C₂×C₆₇	1	2	yes	—
251	Q₈×C₄⋊C₄	C₂×C₆₇	1	2	yes	—
252	C₄×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic
253	C₄×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic
254	C₄×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic
255	C₆₇	#59828	1	1	yes	characteristic, direct product factor, Sylow
256	#557	—	2	16	no	—
257	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
258	C₂×C₄⋊Q₈	C₂²×C₆₇	1	4	yes	—
259	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
260	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
261	C₂×#125	C₂²×C₆₇	1	4	yes	—
262	C₂×#125	C₂²×C₆₇	1	8	yes	—
263	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
264	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
265	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
266	C₂×C₄×Q₈	C₂²×C₆₇	1	4	yes	—
267	C₂×C₄×Q₈	C₂²×C₆₇	1	4	yes	—
268	#400	C₂²×C₆₇	1	8	yes	—
269	#398	C₂²×C₆₇	1	8	yes	—
270	#398	C₂²×C₆₇	1	8	yes	—
271	#398	C₂²×C₆₇	1	8	yes	—
272	#391	C₂²×C₆₇	1	2	yes	—
273	#391	C₂²×C₆₇	1	2	yes	—
274	#391	C₂²×C₆₇	1	2	yes	—
275	#391	C₂²×C₆₇	1	4	yes	—
276	#391	C₂²×C₆₇	1	4	yes	—
277	#389	C₂²×C₆₇	1	2	yes	—
278	#389	C₂²×C₆₇	1	2	yes	—
279	#389	C₂²×C₆₇	1	2	yes	—
280	#389	C₂²×C₆₇	1	2	yes	—
281	#389	C₂²×C₆₇	1	2	yes	—
282	#389	C₂²×C₆₇	1	2	yes	—
283	#389	C₂²×C₆₇	1	4	yes	—
284	#389	C₂²×C₆₇	1	4	yes	—
285	#389	C₂²×C₆₇	1	8	yes	—
286	#389	C₂²×C₆₇	1	4	yes	—
287	#389	C₂²×C₆₇	1	4	yes	—
288	#387	C₂²×C₆₇	1	8	yes	—
289	#387	C₂²×C₆₇	1	8	yes	—
290	#384	C₂²×C₆₇	1	1	yes	characteristic
291	#384	C₂²×C₆₇	1	1	yes	characteristic
292	#384	C₂²×C₆₇	1	1	yes	characteristic
293	#384	C₂²×C₆₇	1	4	yes	—
294	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
295	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
296	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
297	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
298	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
299	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
300	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
301	C₄³	C₂²×C₆₇	1	1	yes	characteristic
302	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
303	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
304	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
305	C₄⋊Q₈	—	2	16	no	—
306	C₄⋊Q₈	—	2	4	no	—
307	C₄⋊Q₈	D₄×C₆₇	1	8	yes	—
308	C₄⋊Q₈	Q₈×C₆₇	1	8	yes	complemented
309	#125	—	2	32	no	—
310	#125	—	2	16	no	—
311	C₄×Q₈	—	2	16	no	—
312	C₄×Q₈	—	2	16	no	—
313	C₄×Q₈	—	2	8	no	—
314	C₄×Q₈	—	2	8	no	—
315	C₄×Q₈	—	2	8	no	—
316	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
317	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
318	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
319	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
320	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
321	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
322	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
323	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
324	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
325	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
326	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
327	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
328	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
329	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
330	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
331	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
332	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
333	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
334	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
335	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
336	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
337	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
338	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
339	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
340	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
341	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
342	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
343	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
344	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
345	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
346	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
347	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
348	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
349	#95	C₂³×C₆₇	1	4	yes	—
350	#95	C₂³×C₆₇	1	4	yes	—
351	#95	C₂³×C₆₇	1	4	yes	—
352	#95	C₂³×C₆₇	1	4	yes	—
353	#95	C₂³×C₆₇	1	4	yes	—
354	#95	C₂³×C₆₇	1	8	yes	—
355	#95	C₂³×C₆₇	1	4	yes	—
356	#95	C₂³×C₆₇	1	8	yes	—
357	#95	C₂³×C₆₇	1	8	yes	—
358	C₂×Q₈	—	2	4	no	—
359	C₂×Q₈	—	2	4	no	—
360	C₂×Q₈	—	2	4	no	—
361	C₂×Q₈	—	2	4	no	—
362	C₂×Q₈	—	2	4	no	—
363	C₂×Q₈	—	2	4	no	—
364	C₂×Q₈	—	2	4	no	—
365	C₂×Q₈	C₂×D₄×C₆₇	1	8	yes	—
366	C₂×Q₈	C₂×Q₈×C₆₇	1	8	yes	—
367	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
368	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
369	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
370	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
371	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
372	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
373	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
374	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
375	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
376	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
377	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
378	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
379	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
380	C₄⋊C₄	—	2	4	no	—
381	C₄⋊C₄	—	2	8	no	—
382	C₄⋊C₄	—	2	16	no	—
383	C₄⋊C₄	—	2	8	no	—
384	C₄⋊C₄	—	2	4	no	—
385	C₄⋊C₄	—	2	16	no	—
386	C₄⋊C₄	—	2	16	no	—
387	C₄⋊C₄	—	2	8	no	—
388	C₄⋊C₄	—	2	4	no	—
389	C₄⋊C₄	—	2	4	no	—
390	C₄⋊C₄	—	2	4	no	—
391	C₄⋊C₄	—	2	4	no	—
392	C₄⋊C₄	—	2	4	no	—
393	C₄⋊C₄	—	2	4	no	—
394	C₄⋊C₄	—	4	2	no	—
395	C₄⋊C₄	—	4	2	no	—
396	C₄⋊C₄	—	4	2	no	—
397	C₄⋊C₄	—	4	8	no	—
398	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
399	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
400	C₄⋊C₄	C₂×Q₈×C₆₇	1	8	yes	—
401	C₄⋊C₄	C₂×Q₈×C₆₇	1	8	yes	—
402	C₄²	—	2	16	no	—
403	C₄²	—	2	8	no	—
404	C₄²	—	2	8	no	—
405	C₄²	—	2	8	no	—
406	C₄²	—	2	2	no	—
407	C₄²	—	2	2	no	—
408	C₄²	—	2	2	no	—
409	C₄²	—	4	1	no	—
410	C₄²	C₂×D₄×C₆₇	1	4	yes	—
411	C₄²	C₂×Q₈×C₆₇	1	4	yes	—
412	C₄²	#41×C₆₇	1	4	yes	—
413	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
414	Q₈	—	2	16	no	—
415	Q₈	—	2	16	no	—
416	Q₈	—	4	8	no	—
417	C₂×C₄	—	2	4	no	—
418	C₂×C₄	—	2	4	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	4	no	—
423	C₂×C₄	—	2	4	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	4	no	—
428	C₂×C₄	—	2	4	no	—
429	C₂×C₄	—	2	4	no	—
430	C₂×C₄	—	2	4	no	—
431	C₂×C₄	—	2	4	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	4	no	—
436	C₂×C₄	—	2	4	no	—
437	C₂×C₄	—	2	4	no	—
438	C₂×C₄	—	2	4	no	—
439	C₂×C₄	—	2	4	no	—
440	C₂×C₄	—	2	4	no	—
441	C₂×C₄	—	2	4	no	—
442	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
443	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
444	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
445	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
446	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
447	C₂×C₄	C₂²×Q₈×C₆₇	1	8	yes	—
448	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
449	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
450	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
451	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
452	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
453	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
454	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
455	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
456	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
457	C₂²	C₂³×Q₈×C₆₇	1	1	yes	characteristic
458	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
459	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
460	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
461	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
462	C₂²	#585×C₆₇	1	1	yes	characteristic
463	C₄	—	2	8	no	—
464	C₄	—	2	8	no	—
465	C₄	—	2	2	no	—
466	C₄	—	2	2	no	—
467	C₄	—	2	2	no	—
468	C₄	—	4	1	no	—
469	C₄	—	4	4	no	—
470	C₄	—	4	4	no	—
471	C₄	—	4	4	no	—
472	C₄	—	4	4	no	—
473	C₄	#547×C₆₇	1	4	yes	—
474	C₄	D₄×Q₈×C₆₇	1	4	yes	—
475	C₄	#557×C₆₇	1	4	yes	complemented
476	C₂	C₂×#547×C₆₇	1	1	yes	characteristic
477	C₂	C₂×D₄×Q₈×C₆₇	1	1	yes	characteristic
478	C₂	C₂×#557×C₆₇	1	1	yes	characteristic
479	C₂	C₆₇×#3497	1	1	yes	characteristic
480	C₂	C₆₇×#3502	1	1	yes	characteristic
481	C₂	C₆₇×#3507	1	1	yes	characteristic
482	C₂	C₆₇×#3556	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
66. C₄⋊Q₈×C₆₇	Q₈	semidirect
233. C₄×C₆₇	#557	semidirect
234. #59828	C₆₇	direct (N × H)
255. C₆₇	#59828	direct (N × H)
308. C₄⋊Q₈	Q₈×C₆₇	semidirect
475. C₄	#557×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#557×C₆₇	2
C₆₇×#2726	2
C₆₇×#2582	2
C₆₇×#2552	2
C₆₇×#2546	2
C₆₇×#2506	2
C₆₇×#2453	2
C₆₇×#2446	2
C₆₇×#2365	2
Q₈×C₄⋊C₄×C₆₇	2
C₄×C₄⋊Q₈×C₆₇	2
#59828	67

Subgroups — C67×#59828 #30739079

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#59828 #30739079