Subgroups — C₆₇×#32906 #30712157

Order 17152.

1436 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#462×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×#451×C₆₇	C₂	1	2	yes	maximal
4	C₆₇×#2857	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1878	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1872	C₂	1	2	yes	maximal
7	C₆₇×#1803	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1797	C₂	1	1	yes	maximal, characteristic
9	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×Q₁₆×C₆₇	C₂²	1	4	yes	—
11	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄×Q₈×C₆₇	—	2	8	no	—
14	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
15	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
16	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
18	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
19	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₆₇	—	2	1	no	—
21	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
22	#462×C₆₇	—	2	8	no	—
23	#451×C₆₇	—	2	8	no	—
24	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
25	C₂×#103×C₆₇	C₂²	1	4	yes	—
26	C₂×#103×C₆₇	C₂²	1	4	yes	—
27	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
28	#391×C₆₇	C₂²	1	2	yes	—
29	#389×C₆₇	—	2	1	no	—
30	#389×C₆₇	—	2	1	no	—
31	#384×C₆₇	C₂²	1	1	yes	characteristic
32	#380×C₆₇	—	2	1	no	—
33	#380×C₆₇	—	2	1	no	—
34	#340×C₆₇	C₂²	1	4	yes	—
35	#336×C₆₇	C₂²	1	2	yes	—
36	C₂²×Q₈×C₆₇	—	2	4	no	—
37	C₂²×Q₈×C₆₇	—	2	4	no	—
38	C₂²×Q₈×C₆₇	—	2	1	no	—
39	C₂²×Q₈×C₆₇	—	2	2	no	—
40	C₂²×Q₈×C₆₇	D₄	1	4	yes	—
41	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
42	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
43	C₂³×C₄×C₆₇	—	2	1	no	—
44	C₂³×C₄×C₆₇	—	2	1	no	—
45	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
46	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
48	C₂×Q₁₆×C₆₇	—	2	8	no	—
49	C₂×Q₁₆×C₆₇	—	2	8	no	—
50	C₂×Q₁₆×C₆₇	—	2	8	no	—
51	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
52	C₄⋊Q₈×C₆₇	—	2	4	no	—
53	C₂²⋊Q₈×C₆₇	—	2	4	no	—
54	C₂²⋊Q₈×C₆₇	—	2	4	no	—
55	C₄×Q₈×C₆₇	—	2	8	no	—
56	C₄×Q₈×C₆₇	—	2	8	no	—
57	C₄×Q₈×C₆₇	—	2	4	no	—
58	C₄×Q₈×C₆₇	—	2	4	no	—
59	C₄×Q₈×C₆₇	—	2	4	no	—
60	C₄×Q₈×C₆₇	D₄	1	8	yes	—
61	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
63	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
64	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
65	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
66	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
67	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
68	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
69	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
70	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
71	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
72	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
73	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
74	C₂×C₄²×C₆₇	—	2	8	no	—
75	C₂×C₄²×C₆₇	—	2	2	no	—
76	C₂×C₄²×C₆₇	—	2	2	no	—
77	C₂×C₄²×C₆₇	—	2	4	no	—
78	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
79	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
80	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
81	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
82	C₄⋊C₈×C₆₇	—	2	4	no	—
83	#103×C₆₇	—	2	8	no	—
84	#103×C₆₇	—	2	8	no	—
85	C₂²⋊C₈×C₆₇	—	2	4	no	—
86	#95×C₆₇	—	2	2	no	—
87	#95×C₆₇	—	2	2	no	—
88	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₂×Q₈×C₆₇	—	2	4	no	—
90	C₂×Q₈×C₆₇	—	2	16	no	—
91	C₂×Q₈×C₆₇	—	2	4	no	—
92	C₂×Q₈×C₆₇	—	2	16	no	—
93	C₂×Q₈×C₆₇	—	2	4	no	—
94	C₂×Q₈×C₆₇	—	2	2	no	—
95	C₂×Q₈×C₆₇	—	2	2	no	—
96	C₂×Q₈×C₆₇	—	2	2	no	—
97	C₂×Q₈×C₆₇	—	2	4	no	—
98	C₂×Q₈×C₆₇	—	2	2	no	—
99	C₂×Q₈×C₆₇	—	2	2	no	—
100	C₂×Q₈×C₆₇	—	2	4	no	—
101	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
102	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
103	C₂×Q₈×C₆₇	#41	1	4	yes	—
104	C₂²×C₄×C₆₇	—	2	4	no	—
105	C₂²×C₄×C₆₇	—	2	4	no	—
106	C₂²×C₄×C₆₇	—	2	4	no	—
107	C₂²×C₄×C₆₇	—	2	2	no	—
108	C₂²×C₄×C₆₇	—	2	2	no	—
109	C₂²×C₄×C₆₇	—	2	2	no	—
110	C₂²×C₄×C₆₇	—	2	2	no	—
111	C₂²×C₄×C₆₇	—	2	2	no	—
112	C₂²×C₄×C₆₇	—	2	4	no	—
113	C₂²×C₄×C₆₇	—	2	2	no	—
114	C₂²×C₄×C₆₇	—	2	2	no	—
115	C₂²×C₄×C₆₇	—	2	2	no	—
116	C₂²×C₄×C₆₇	—	2	2	no	—
117	C₂²×C₄×C₆₇	—	2	2	no	—
118	C₂²×C₄×C₆₇	—	4	1	no	—
119	C₂²×C₄×C₆₇	—	4	2	no	—
120	C₂²×C₄×C₆₇	—	4	2	no	—
121	C₂²×C₄×C₆₇	Q₁₆	1	4	yes	complemented
122	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
126	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₆₇	#41	1	2	yes	—
129	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
130	Q₁₆×C₆₇	—	4	16	no	—
131	C₂×C₈×C₆₇	—	2	4	no	—
132	C₂×C₈×C₆₇	—	2	4	no	—
133	C₂×C₈×C₆₇	—	2	4	no	—
134	C₄⋊C₄×C₆₇	—	2	8	no	—
135	C₄⋊C₄×C₆₇	—	2	4	no	—
136	C₄⋊C₄×C₆₇	—	2	8	no	—
137	C₄⋊C₄×C₆₇	—	2	2	no	—
138	C₄⋊C₄×C₆₇	—	2	2	no	—
139	C₄⋊C₄×C₆₇	—	2	2	no	—
140	C₄⋊C₄×C₆₇	—	4	2	no	—
141	C₄⋊C₄×C₆₇	—	4	2	no	—
142	C₄⋊C₄×C₆₇	—	4	2	no	—
143	C₄⋊C₄×C₆₇	—	4	4	no	—
144	C₄⋊C₄×C₆₇	—	4	2	no	—
145	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
146	C₂²⋊C₄×C₆₇	—	4	4	no	—
147	C₄²×C₆₇	—	2	8	no	—
148	C₄²×C₆₇	—	2	8	no	—
149	C₄²×C₆₇	—	2	2	no	—
150	C₄²×C₆₇	—	2	2	no	—
151	C₄²×C₆₇	—	2	2	no	—
152	C₄²×C₆₇	C₂×D₄	1	4	yes	—
153	C₂³×C₆₇	—	2	1	no	—
154	C₂³×C₆₇	—	2	1	no	—
155	C₂³×C₆₇	—	2	1	no	—
156	C₂³×C₆₇	—	2	1	no	—
157	C₂³×C₆₇	#121	1	2	yes	—
158	C₂³×C₆₇	C₂×Q₁₆	1	2	yes	—
159	C₂³×C₆₇	#137	1	2	yes	—
160	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
161	Q₈×C₆₇	—	2	16	no	—
162	Q₈×C₆₇	—	2	4	no	—
163	Q₈×C₆₇	—	2	4	no	—
164	Q₈×C₆₇	—	2	4	no	—
165	Q₈×C₆₇	—	4	4	no	—
166	Q₈×C₆₇	—	4	4	no	—
167	Q₈×C₆₇	#121	1	8	yes	—
168	C₂×C₄×C₆₇	—	2	16	no	—
169	C₂×C₄×C₆₇	—	2	1	no	—
170	C₂×C₄×C₆₇	—	2	4	no	—
171	C₂×C₄×C₆₇	—	2	1	no	—
172	C₂×C₄×C₆₇	—	2	4	no	—
173	C₂×C₄×C₆₇	—	2	4	no	—
174	C₂×C₄×C₆₇	—	2	1	no	—
175	C₂×C₄×C₆₇	—	2	1	no	—
176	C₂×C₄×C₆₇	—	2	1	no	—
177	C₂×C₄×C₆₇	—	2	1	no	—
178	C₂×C₄×C₆₇	—	2	4	no	—
179	C₂×C₄×C₆₇	—	2	4	no	—
180	C₂×C₄×C₆₇	—	2	8	no	—
181	C₂×C₄×C₆₇	—	2	1	no	—
182	C₂×C₄×C₆₇	—	2	4	no	—
183	C₂×C₄×C₆₇	—	2	1	no	—
184	C₂×C₄×C₆₇	—	2	1	no	—
185	C₂×C₄×C₆₇	—	4	4	no	—
186	C₂×C₄×C₆₇	—	4	8	no	—
187	C₂×C₄×C₆₇	—	4	2	no	—
188	C₂×C₄×C₆₇	—	4	2	no	—
189	C₂×C₄×C₆₇	—	4	2	no	—
190	C₂×C₄×C₆₇	—	4	2	no	—
191	C₂×C₄×C₆₇	—	4	2	no	—
192	C₂×C₄×C₆₇	—	4	2	no	—
193	C₂×C₄×C₆₇	—	4	2	no	—
194	C₂×C₄×C₆₇	—	4	2	no	—
195	C₂×C₄×C₆₇	#120	1	2	yes	—
196	C₂×C₄×C₆₇	#120	1	2	yes	—
197	C₂×C₄×C₆₇	#121	1	2	yes	—
198	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
199	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
200	C₂×C₄×C₆₇	#137	1	2	yes	—
201	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
202	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
203	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
204	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
205	C₈×C₆₇	—	4	4	no	—
206	C₂²×C₆₇	—	2	2	no	—
207	C₂²×C₆₇	—	2	2	no	—
208	C₂²×C₆₇	—	2	2	no	—
209	C₂²×C₆₇	—	2	2	no	—
210	C₂²×C₆₇	—	2	2	no	—
211	C₂²×C₆₇	—	2	2	no	—
212	C₂²×C₆₇	#462	1	4	yes	complemented
213	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
214	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
215	C₂²×C₆₇	#525	1	1	yes	characteristic
216	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
217	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
218	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
219	C₂²×C₆₇	#575	1	1	yes	characteristic
220	C₄×C₆₇	—	2	8	no	—
221	C₄×C₆₇	—	2	2	no	—
222	C₄×C₆₇	—	2	2	no	—
223	C₄×C₆₇	—	2	2	no	—
224	C₄×C₆₇	—	4	1	no	—
225	C₄×C₆₇	—	4	4	no	—
226	C₄×C₆₇	—	8	2	no	—
227	C₄×C₆₇	—	8	2	no	—
228	C₄×C₆₇	#451	1	4	yes	complemented
229	C₄×C₆₇	C₂×#121	1	2	yes	—
230	C₄×C₆₇	#525	1	2	yes	—
231	#32906	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
232	C₂×C₆₇	—	2	4	no	—
233	C₂×C₆₇	#2847	1	1	yes	characteristic
234	C₂×C₆₇	C₂×#451	1	1	yes	characteristic
235	C₂×C₆₇	#3007	1	1	yes	characteristic
236	C₂×C₆₇	C₂×#462	1	1	yes	characteristic
237	C₂×C₆₇	#3037	1	1	yes	characteristic
238	C₂×C₆₇	#3104	1	1	yes	characteristic
239	C₂×C₆₇	#3113	1	1	yes	characteristic
240	C₂²×C₄×Q₈	C₂×C₆₇	1	1	yes	characteristic
241	C₂×#462	C₂×C₆₇	1	2	yes	complemented
242	C₂×#451	C₂×C₆₇	1	2	yes	—
243	#2857	C₂×C₆₇	1	1	yes	characteristic
244	#1878	C₂×C₆₇	1	4	yes	complemented
245	#1872	C₂×C₆₇	1	2	yes	—
246	#1803	C₂×C₆₇	1	2	yes	complemented
247	#1797	C₂×C₆₇	1	1	yes	characteristic
248	C₆₇	#32906	1	1	yes	characteristic, direct product factor, Sylow
249	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
250	C₂²×Q₁₆	C₂²×C₆₇	1	4	yes	—
251	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
252	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
253	C₂×C₄×Q₈	—	2	8	no	—
254	C₂×C₄×Q₈	—	2	1	no	—
255	C₂×C₄×Q₈	—	2	1	no	—
256	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
257	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
258	C₂²×C₄⋊C₄	—	2	1	no	—
259	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
260	C₂²×C₄²	—	2	1	no	—
261	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
262	#462	—	2	8	no	—
263	#451	—	2	8	no	—
264	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
265	C₂×#103	C₂²×C₆₇	1	4	yes	—
266	C₂×#103	C₂²×C₆₇	1	4	yes	—
267	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
268	#391	C₂²×C₆₇	1	2	yes	—
269	#389	—	2	1	no	—
270	#389	—	2	1	no	—
271	#384	C₂²×C₆₇	1	1	yes	characteristic
272	#380	—	2	1	no	—
273	#380	—	2	1	no	—
274	#340	C₂²×C₆₇	1	4	yes	—
275	#336	C₂²×C₆₇	1	2	yes	—
276	C₂²×Q₈	—	2	4	no	—
277	C₂²×Q₈	—	2	4	no	—
278	C₂²×Q₈	—	2	1	no	—
279	C₂²×Q₈	—	2	2	no	—
280	C₂²×Q₈	D₄×C₆₇	1	4	yes	—
281	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
282	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
283	C₂³×C₄	—	2	1	no	—
284	C₂³×C₄	—	2	1	no	—
285	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
286	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
287	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
288	C₂×Q₁₆	—	2	8	no	—
289	C₂×Q₁₆	—	2	8	no	—
290	C₂×Q₁₆	—	2	8	no	—
291	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
292	C₄⋊Q₈	—	2	4	no	—
293	C₂²⋊Q₈	—	2	4	no	—
294	C₂²⋊Q₈	—	2	4	no	—
295	C₄×Q₈	—	2	8	no	—
296	C₄×Q₈	—	2	8	no	—
297	C₄×Q₈	—	2	4	no	—
298	C₄×Q₈	—	2	4	no	—
299	C₄×Q₈	—	2	4	no	—
300	C₄×Q₈	D₄×C₆₇	1	8	yes	—
301	C₂×C₄⋊C₄	—	2	8	no	—
302	C₂×C₄⋊C₄	—	2	2	no	—
303	C₂×C₄⋊C₄	—	2	2	no	—
304	C₂×C₄⋊C₄	—	2	1	no	—
305	C₂×C₄⋊C₄	—	2	4	no	—
306	C₂×C₄⋊C₄	—	2	2	no	—
307	C₂×C₄⋊C₄	—	2	1	no	—
308	C₂×C₄⋊C₄	—	2	1	no	—
309	C₂×C₄⋊C₄	—	2	1	no	—
310	C₂×C₄⋊C₄	—	2	1	no	—
311	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
312	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
313	C₂×C₂²⋊C₄	—	2	2	no	—
314	C₂×C₄²	—	2	8	no	—
315	C₂×C₄²	—	2	2	no	—
316	C₂×C₄²	—	2	2	no	—
317	C₂×C₄²	—	2	4	no	—
318	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
319	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
320	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
321	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
322	C₄⋊C₈	—	2	4	no	—
323	#103	—	2	8	no	—
324	#103	—	2	8	no	—
325	C₂²⋊C₈	—	2	4	no	—
326	#95	—	2	2	no	—
327	#95	—	2	2	no	—
328	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
329	C₂×Q₈	—	2	4	no	—
330	C₂×Q₈	—	2	16	no	—
331	C₂×Q₈	—	2	4	no	—
332	C₂×Q₈	—	2	16	no	—
333	C₂×Q₈	—	2	4	no	—
334	C₂×Q₈	—	2	2	no	—
335	C₂×Q₈	—	2	2	no	—
336	C₂×Q₈	—	2	2	no	—
337	C₂×Q₈	—	2	4	no	—
338	C₂×Q₈	—	2	2	no	—
339	C₂×Q₈	—	2	2	no	—
340	C₂×Q₈	—	2	4	no	—
341	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
342	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
343	C₂×Q₈	#41×C₆₇	1	4	yes	—
344	C₂²×C₄	—	2	4	no	—
345	C₂²×C₄	—	2	4	no	—
346	C₂²×C₄	—	2	4	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	—	2	2	no	—
349	C₂²×C₄	—	2	2	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	2	2	no	—
352	C₂²×C₄	—	2	4	no	—
353	C₂²×C₄	—	2	2	no	—
354	C₂²×C₄	—	2	2	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	2	2	no	—
357	C₂²×C₄	—	2	2	no	—
358	C₂²×C₄	—	4	1	no	—
359	C₂²×C₄	—	4	2	no	—
360	C₂²×C₄	—	4	2	no	—
361	C₂²×C₄	Q₁₆×C₆₇	1	4	yes	complemented
362	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
363	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
364	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
365	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
366	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
367	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
368	C₂²×C₄	#41×C₆₇	1	2	yes	—
369	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
370	Q₁₆	—	4	16	no	—
371	C₂×C₈	—	2	4	no	—
372	C₂×C₈	—	2	4	no	—
373	C₂×C₈	—	2	4	no	—
374	C₄⋊C₄	—	2	8	no	—
375	C₄⋊C₄	—	2	4	no	—
376	C₄⋊C₄	—	2	8	no	—
377	C₄⋊C₄	—	2	2	no	—
378	C₄⋊C₄	—	2	2	no	—
379	C₄⋊C₄	—	2	2	no	—
380	C₄⋊C₄	—	4	2	no	—
381	C₄⋊C₄	—	4	2	no	—
382	C₄⋊C₄	—	4	2	no	—
383	C₄⋊C₄	—	4	4	no	—
384	C₄⋊C₄	—	4	2	no	—
385	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
386	C₂²⋊C₄	—	4	4	no	—
387	C₄²	—	2	8	no	—
388	C₄²	—	2	8	no	—
389	C₄²	—	2	2	no	—
390	C₄²	—	2	2	no	—
391	C₄²	—	2	2	no	—
392	C₄²	C₂×D₄×C₆₇	1	4	yes	—
393	C₂³	—	2	1	no	—
394	C₂³	—	2	1	no	—
395	C₂³	—	2	1	no	—
396	C₂³	—	2	1	no	—
397	C₂³	#121×C₆₇	1	2	yes	—
398	C₂³	C₂×Q₁₆×C₆₇	1	2	yes	—
399	C₂³	#137×C₆₇	1	2	yes	—
400	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
401	Q₈	—	2	16	no	—
402	Q₈	—	2	4	no	—
403	Q₈	—	2	4	no	—
404	Q₈	—	2	4	no	—
405	Q₈	—	4	4	no	—
406	Q₈	—	4	4	no	—
407	Q₈	#121×C₆₇	1	8	yes	—
408	C₂×C₄	—	2	16	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	4	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	4	no	—
413	C₂×C₄	—	2	4	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	8	no	—
419	C₂×C₄	—	2	4	no	—
420	C₂×C₄	—	2	4	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	4	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	4	4	no	—
426	C₂×C₄	—	4	8	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	2	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	2	no	—
435	C₂×C₄	#120×C₆₇	1	2	yes	—
436	C₂×C₄	#120×C₆₇	1	2	yes	—
437	C₂×C₄	#121×C₆₇	1	2	yes	—
438	C₂×C₄	C₂×Q₁₆×C₆₇	1	2	yes	—
439	C₂×C₄	C₂×Q₁₆×C₆₇	1	2	yes	—
440	C₂×C₄	#137×C₆₇	1	2	yes	—
441	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
442	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
443	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
444	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
445	C₈	—	4	4	no	—
446	C₂²	—	2	2	no	—
447	C₂²	—	2	2	no	—
448	C₂²	—	2	2	no	—
449	C₂²	—	2	2	no	—
450	C₂²	—	2	2	no	—
451	C₂²	—	2	2	no	—
452	C₂²	#462×C₆₇	1	4	yes	complemented
453	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
454	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
455	C₂²	#525×C₆₇	1	1	yes	characteristic
456	C₂²	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
457	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
458	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
459	C₂²	#575×C₆₇	1	1	yes	characteristic
460	C₄	—	2	8	no	—
461	C₄	—	2	2	no	—
462	C₄	—	2	2	no	—
463	C₄	—	2	2	no	—
464	C₄	—	4	1	no	—
465	C₄	—	4	4	no	—
466	C₄	—	8	2	no	—
467	C₄	—	8	2	no	—
468	C₄	#451×C₆₇	1	4	yes	complemented
469	C₄	C₂×#121×C₆₇	1	2	yes	—
470	C₄	#525×C₆₇	1	2	yes	—
471	C₂	—	2	4	no	—
472	C₂	C₆₇×#2847	1	1	yes	characteristic
473	C₂	C₂×#451×C₆₇	1	1	yes	characteristic
474	C₂	C₆₇×#3007	1	1	yes	characteristic
475	C₂	C₂×#462×C₆₇	1	1	yes	characteristic
476	C₂	C₆₇×#3037	1	1	yes	characteristic
477	C₂	C₆₇×#3104	1	1	yes	characteristic
478	C₂	C₆₇×#3113	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#462×C₆₇	C₂	semidirect
5. C₆₇×#1878	C₂	semidirect
7. C₆₇×#1803	C₂	semidirect
121. C₂²×C₄×C₆₇	Q₁₆	semidirect
212. C₂²×C₆₇	#462	semidirect
228. C₄×C₆₇	#451	semidirect
231. #32906	C₆₇	direct (N × H)
241. C₂×#462	C₂×C₆₇	semidirect
244. #1878	C₂×C₆₇	semidirect
246. #1803	C₂×C₆₇	semidirect
248. C₆₇	#32906	direct (N × H)
361. C₂²×C₄	Q₁₆×C₆₇	semidirect
452. C₂²	#462×C₆₇	semidirect
468. C₄	#451×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×Q₈×C₆₇	2
C₂×#462×C₆₇	2
C₂×#451×C₆₇	2
C₆₇×#2857	2
C₆₇×#1878	2
C₆₇×#1872	2
C₆₇×#1803	2
C₆₇×#1797	2
#32906	67

Subgroups — C67×#32906 #30712157

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32906 #30712157