Subgroups — C₆₇×#21873 #30701124

Order 17152.

1022 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#541×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#433×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2317	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1992	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1991	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1960	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1956	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1873	C₂	1	2	yes	maximal
9	C₆₇×#1872	C₂	1	2	yes	maximal
10	C₆₇×#1805	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1803	C₂	1	2	yes	maximal, complemented
12	#541×C₆₇	—	2	4	no	—
13	#541×C₆₇	C₄	1	8	yes	complemented
14	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#126×C₆₇	—	2	2	no	—
16	C₂×#125×C₆₇	—	2	2	no	—
17	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
20	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
23	#433×C₆₇	—	2	4	no	—
24	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
25	C₂×#103×C₆₇	C₂²	1	2	yes	—
26	C₂×#103×C₆₇	C₂²	1	2	yes	—
27	C₂×#103×C₆₇	C₂²	1	2	yes	—
28	C₂×#103×C₆₇	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
30	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	#389×C₆₇	—	2	2	no	—
33	#385×C₆₇	—	2	2	no	—
34	#384×C₆₇	C₂²	1	1	yes	characteristic
35	#384×C₆₇	C₂²	1	1	yes	characteristic
36	#382×C₆₇	C₂²	1	1	yes	characteristic
37	#382×C₆₇	C₂²	1	1	yes	characteristic
38	#340×C₆₇	C₂²	1	2	yes	—
39	#340×C₆₇	C₂²	1	2	yes	—
40	#340×C₆₇	C₂²	1	2	yes	—
41	#340×C₆₇	C₂²	1	2	yes	—
42	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
43	C₂³×C₄×C₆₇	—	2	1	no	—
44	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
46	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
47	C₄⋊Q₈×C₆₇	—	2	2	no	—
48	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	complemented
49	#126×C₆₇	—	2	16	no	—
50	#125×C₆₇	—	2	16	no	—
51	#125×C₆₇	—	2	2	no	—
52	#125×C₆₇	C₂×C₄	1	4	yes	complemented
53	C₂²⋊Q₈×C₆₇	—	2	4	no	—
54	C₂²⋊Q₈×C₆₇	C₂×C₄	1	8	yes	—
55	C₄×Q₈×C₆₇	—	2	4	no	—
56	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
57	#117×C₆₇	—	4	2	no	—
58	#117×C₆₇	C₂×C₄	1	4	yes	—
59	#117×C₆₇	D₄	1	4	yes	—
60	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
61	C₂×C₄⋊C₄×C₆₇	—	2	2	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	2	no	—
66	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
67	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
68	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
74	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
75	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
76	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
77	C₂×C₄²×C₆₇	—	2	2	no	—
78	C₂×C₄²×C₆₇	—	2	1	no	—
79	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
80	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
81	C₄⋊C₈×C₆₇	—	2	4	no	—
82	#103×C₆₇	—	4	2	no	—
83	#103×C₆₇	—	4	2	no	—
84	#103×C₆₇	—	4	2	no	—
85	#103×C₆₇	—	4	2	no	—
86	C₂²⋊C₈×C₆₇	—	2	4	no	—
87	#97×C₆₇	—	2	2	no	—
88	C₄×C₈×C₆₇	—	2	2	no	—
89	#95×C₆₇	—	2	2	no	—
90	#95×C₆₇	—	2	2	no	—
91	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₂×Q₈×C₆₇	—	2	2	no	—
93	C₂×Q₈×C₆₇	—	2	2	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	C₂²×C₄	1	4	yes	—
98	C₂²×C₄×C₆₇	—	2	2	no	—
99	C₂²×C₄×C₆₇	—	2	2	no	—
100	C₂²×C₄×C₆₇	—	2	2	no	—
101	C₂²×C₄×C₆₇	—	2	2	no	—
102	C₂²×C₄×C₆₇	—	2	1	no	—
103	C₂²×C₄×C₆₇	—	2	1	no	—
104	C₂²×C₄×C₆₇	—	2	1	no	—
105	C₂²×C₄×C₆₇	—	2	1	no	—
106	C₂²×C₄×C₆₇	—	4	2	no	—
107	C₂²×C₄×C₆₇	—	8	2	no	—
108	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
109	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
113	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
114	C₂×C₈×C₆₇	—	2	2	no	—
115	C₂×C₈×C₆₇	—	2	2	no	—
116	C₂×C₈×C₆₇	—	2	2	no	—
117	C₂×C₈×C₆₇	—	2	2	no	—
118	C₂×C₈×C₆₇	—	2	2	no	—
119	C₂×C₈×C₆₇	—	2	2	no	—
120	C₄⋊C₄×C₆₇	—	2	8	no	—
121	C₄⋊C₄×C₆₇	—	2	8	no	—
122	C₄⋊C₄×C₆₇	—	2	8	no	—
123	C₄⋊C₄×C₆₇	—	2	8	no	—
124	C₄⋊C₄×C₆₇	—	2	2	no	—
125	C₄⋊C₄×C₆₇	—	2	2	no	—
126	C₄⋊C₄×C₆₇	—	2	2	no	—
127	C₄⋊C₄×C₆₇	—	2	2	no	—
128	C₄⋊C₄×C₆₇	—	2	2	no	—
129	C₄⋊C₄×C₆₇	—	4	2	no	—
130	C₄⋊C₄×C₆₇	—	8	2	no	—
131	C₄⋊C₄×C₆₇	—	8	2	no	—
132	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
133	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
134	C₂²⋊C₄×C₆₇	—	2	8	no	—
135	C₂²⋊C₄×C₆₇	—	4	2	no	—
136	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
137	C₄²×C₆₇	—	2	8	no	—
138	C₄²×C₆₇	—	4	2	no	—
139	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
140	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
141	C₄²×C₆₇	C₂×D₄	1	2	yes	—
142	C₄²×C₆₇	C₂×D₄	1	2	yes	—
143	C₂³×C₆₇	—	2	1	no	—
144	C₂³×C₆₇	—	2	1	no	—
145	C₂³×C₆₇	—	2	2	no	—
146	C₂³×C₆₇	—	2	2	no	—
147	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
148	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
149	Q₈×C₆₇	—	2	4	no	—
150	Q₈×C₆₇	—	2	4	no	—
151	Q₈×C₆₇	—	4	2	no	—
152	Q₈×C₆₇	—	4	2	no	—
153	C₂×C₄×C₆₇	—	2	1	no	—
154	C₂×C₄×C₆₇	—	2	1	no	—
155	C₂×C₄×C₆₇	—	2	2	no	—
156	C₂×C₄×C₆₇	—	2	2	no	—
157	C₂×C₄×C₆₇	—	2	2	no	—
158	C₂×C₄×C₆₇	—	2	2	no	—
159	C₂×C₄×C₆₇	—	2	4	no	—
160	C₂×C₄×C₆₇	—	2	4	no	—
161	C₂×C₄×C₆₇	—	2	4	no	—
162	C₂×C₄×C₆₇	—	2	4	no	—
163	C₂×C₄×C₆₇	—	4	2	no	—
164	C₂×C₄×C₆₇	—	4	2	no	—
165	C₂×C₄×C₆₇	—	4	2	no	—
166	C₂×C₄×C₆₇	—	4	2	no	—
167	C₂×C₄×C₆₇	—	4	1	no	—
168	C₂×C₄×C₆₇	—	4	1	no	—
169	C₂×C₄×C₆₇	—	4	1	no	—
170	C₂×C₄×C₆₇	—	4	1	no	—
171	C₂×C₄×C₆₇	—	4	2	no	—
172	C₂×C₄×C₆₇	—	4	2	no	—
173	C₂×C₄×C₆₇	—	4	2	no	—
174	C₂×C₄×C₆₇	—	4	2	no	—
175	C₂×C₄×C₆₇	—	8	4	no	—
176	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
177	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
178	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
179	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
180	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
181	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
182	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
183	C₈×C₆₇	—	4	2	no	—
184	C₈×C₆₇	—	4	2	no	—
185	C₂²×C₆₇	—	2	2	no	—
186	C₂²×C₆₇	—	2	4	no	—
187	C₂²×C₆₇	—	4	1	no	—
188	C₂²×C₆₇	—	4	1	no	—
189	C₂²×C₆₇	—	4	2	no	—
190	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
191	C₂²×C₆₇	#535	1	1	yes	characteristic
192	C₂²×C₆₇	#536	1	1	yes	characteristic
193	C₂²×C₆₇	#577	1	2	yes	—
194	C₂²×C₆₇	#578	1	2	yes	—
195	C₄×C₆₇	—	2	2	no	—
196	C₄×C₆₇	—	2	4	no	—
197	C₄×C₆₇	—	2	4	no	—
198	C₄×C₆₇	—	4	4	no	—
199	C₄×C₆₇	—	4	4	no	—
200	C₄×C₆₇	—	8	1	no	—
201	C₄×C₆₇	—	8	1	no	—
202	C₄×C₆₇	—	8	1	no	—
203	C₄×C₆₇	—	8	1	no	—
204	C₄×C₆₇	#520	1	2	yes	—
205	C₄×C₆₇	#537	1	2	yes	—
206	#21873	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
207	C₂×C₆₇	—	4	2	no	—
208	C₂×C₆₇	#2311	1	1	yes	characteristic
209	C₂×C₆₇	#2898	1	2	yes	—
210	C₂×C₆₇	#3222	1	1	yes	characteristic
211	C₂×C₆₇	#3223	1	1	yes	characteristic
212	C₂×C₆₇	#3242	1	2	yes	—
213	C₂×#541	C₂×C₆₇	1	1	yes	characteristic
214	C₂×#433	C₂×C₆₇	1	1	yes	characteristic
215	#2317	C₂×C₆₇	1	1	yes	characteristic
216	#1992	C₂×C₆₇	1	1	yes	characteristic, complemented
217	#1991	C₂×C₆₇	1	1	yes	characteristic, complemented
218	#1960	C₂×C₆₇	1	1	yes	characteristic, complemented
219	#1956	C₂×C₆₇	1	1	yes	characteristic, complemented
220	#1873	C₂×C₆₇	1	2	yes	—
221	#1872	C₂×C₆₇	1	2	yes	—
222	#1805	C₂×C₆₇	1	2	yes	complemented
223	#1803	C₂×C₆₇	1	2	yes	complemented
224	C₆₇	#21873	1	1	yes	characteristic, direct product factor, Sylow
225	#541	—	2	4	no	—
226	#541	C₄×C₆₇	1	8	yes	complemented
227	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
228	C₂×#126	—	2	2	no	—
229	C₂×#125	—	2	2	no	—
230	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
231	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
232	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
233	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
234	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
235	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
236	#433	—	2	4	no	—
237	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
238	C₂×#103	C₂²×C₆₇	1	2	yes	—
239	C₂×#103	C₂²×C₆₇	1	2	yes	—
240	C₂×#103	C₂²×C₆₇	1	2	yes	—
241	C₂×#103	C₂²×C₆₇	1	2	yes	—
242	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
243	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
244	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
245	#389	—	2	2	no	—
246	#385	—	2	2	no	—
247	#384	C₂²×C₆₇	1	1	yes	characteristic
248	#384	C₂²×C₆₇	1	1	yes	characteristic
249	#382	C₂²×C₆₇	1	1	yes	characteristic
250	#382	C₂²×C₆₇	1	1	yes	characteristic
251	#340	C₂²×C₆₇	1	2	yes	—
252	#340	C₂²×C₆₇	1	2	yes	—
253	#340	C₂²×C₆₇	1	2	yes	—
254	#340	C₂²×C₆₇	1	2	yes	—
255	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
256	C₂³×C₄	—	2	1	no	—
257	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
258	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
259	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
260	C₄⋊Q₈	—	2	2	no	—
261	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	complemented
262	#126	—	2	16	no	—
263	#125	—	2	16	no	—
264	#125	—	2	2	no	—
265	#125	C₂×C₄×C₆₇	1	4	yes	complemented
266	C₂²⋊Q₈	—	2	4	no	—
267	C₂²⋊Q₈	C₂×C₄×C₆₇	1	8	yes	—
268	C₄×Q₈	—	2	4	no	—
269	C₄×Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
270	#117	—	4	2	no	—
271	#117	C₂×C₄×C₆₇	1	4	yes	—
272	#117	D₄×C₆₇	1	4	yes	—
273	C₂×C₄⋊C₄	—	2	2	no	—
274	C₂×C₄⋊C₄	—	2	2	no	—
275	C₂×C₄⋊C₄	—	2	2	no	—
276	C₂×C₄⋊C₄	—	2	2	no	—
277	C₂×C₄⋊C₄	—	2	1	no	—
278	C₂×C₄⋊C₄	—	2	2	no	—
279	C₂×C₄⋊C₄	—	2	2	no	—
280	C₂×C₄⋊C₄	—	4	2	no	—
281	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
282	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
283	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
284	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
285	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
286	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
287	C₂×C₂²⋊C₄	—	2	2	no	—
288	C₂×C₂²⋊C₄	—	2	1	no	—
289	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
290	C₂×C₄²	—	2	2	no	—
291	C₂×C₄²	—	2	1	no	—
292	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
293	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
294	C₄⋊C₈	—	2	4	no	—
295	#103	—	4	2	no	—
296	#103	—	4	2	no	—
297	#103	—	4	2	no	—
298	#103	—	4	2	no	—
299	C₂²⋊C₈	—	2	4	no	—
300	#97	—	2	2	no	—
301	C₄×C₈	—	2	2	no	—
302	#95	—	2	2	no	—
303	#95	—	2	2	no	—
304	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
305	C₂×Q₈	—	2	2	no	—
306	C₂×Q₈	—	2	2	no	—
307	C₂×Q₈	—	2	2	no	—
308	C₂×Q₈	—	2	2	no	—
309	C₂×Q₈	—	2	2	no	—
310	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
311	C₂²×C₄	—	2	2	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	—	2	2	no	—
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	1	no	—
317	C₂²×C₄	—	2	1	no	—
318	C₂²×C₄	—	2	1	no	—
319	C₂²×C₄	—	4	2	no	—
320	C₂²×C₄	—	8	2	no	—
321	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
322	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
323	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
324	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
325	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
326	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	2	2	no	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	—	2	2	no	—
333	C₄⋊C₄	—	2	8	no	—
334	C₄⋊C₄	—	2	8	no	—
335	C₄⋊C₄	—	2	8	no	—
336	C₄⋊C₄	—	2	8	no	—
337	C₄⋊C₄	—	2	2	no	—
338	C₄⋊C₄	—	2	2	no	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	4	2	no	—
343	C₄⋊C₄	—	8	2	no	—
344	C₄⋊C₄	—	8	2	no	—
345	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
346	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
347	C₂²⋊C₄	—	2	8	no	—
348	C₂²⋊C₄	—	4	2	no	—
349	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
350	C₄²	—	2	8	no	—
351	C₄²	—	4	2	no	—
352	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
353	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
354	C₄²	C₂×D₄×C₆₇	1	2	yes	—
355	C₄²	C₂×D₄×C₆₇	1	2	yes	—
356	C₂³	—	2	1	no	—
357	C₂³	—	2	1	no	—
358	C₂³	—	2	2	no	—
359	C₂³	—	2	2	no	—
360	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
361	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
362	Q₈	—	2	4	no	—
363	Q₈	—	2	4	no	—
364	Q₈	—	4	2	no	—
365	Q₈	—	4	2	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	4	no	—
373	C₂×C₄	—	2	4	no	—
374	C₂×C₄	—	2	4	no	—
375	C₂×C₄	—	2	4	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	—	4	2	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	4	1	no	—
381	C₂×C₄	—	4	1	no	—
382	C₂×C₄	—	4	1	no	—
383	C₂×C₄	—	4	1	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	—	8	4	no	—
389	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
390	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
391	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
392	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
393	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
394	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
395	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
396	C₈	—	4	2	no	—
397	C₈	—	4	2	no	—
398	C₂²	—	2	2	no	—
399	C₂²	—	2	4	no	—
400	C₂²	—	4	1	no	—
401	C₂²	—	4	1	no	—
402	C₂²	—	4	2	no	—
403	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
404	C₂²	#535×C₆₇	1	1	yes	characteristic
405	C₂²	#536×C₆₇	1	1	yes	characteristic
406	C₂²	#577×C₆₇	1	2	yes	—
407	C₂²	#578×C₆₇	1	2	yes	—
408	C₄	—	2	2	no	—
409	C₄	—	2	4	no	—
410	C₄	—	2	4	no	—
411	C₄	—	4	4	no	—
412	C₄	—	4	4	no	—
413	C₄	—	8	1	no	—
414	C₄	—	8	1	no	—
415	C₄	—	8	1	no	—
416	C₄	—	8	1	no	—
417	C₄	#520×C₆₇	1	2	yes	—
418	C₄	#537×C₆₇	1	2	yes	—
419	C₂	—	4	2	no	—
420	C₂	C₆₇×#2311	1	1	yes	characteristic
421	C₂	C₆₇×#2898	1	2	yes	—
422	C₂	C₆₇×#3222	1	1	yes	characteristic
423	C₂	C₆₇×#3223	1	1	yes	characteristic
424	C₂	C₆₇×#3242	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₇×#1992	C₂	semidirect
5. C₆₇×#1991	C₂	semidirect
6. C₆₇×#1960	C₂	semidirect
7. C₆₇×#1956	C₂	semidirect
10. C₆₇×#1805	C₂	semidirect
11. C₆₇×#1803	C₂	semidirect
13. #541×C₆₇	C₄	semidirect
48. C₄⋊Q₈×C₆₇	C₂×C₄	semidirect
52. #125×C₆₇	C₂×C₄	semidirect
56. C₄×Q₈×C₆₇	C₂×C₄	semidirect
206. #21873	C₆₇	direct (N × H)
216. #1992	C₂×C₆₇	semidirect
217. #1991	C₂×C₆₇	semidirect
218. #1960	C₂×C₆₇	semidirect
219. #1956	C₂×C₆₇	semidirect
222. #1805	C₂×C₆₇	semidirect
223. #1803	C₂×C₆₇	semidirect
224. C₆₇	#21873	direct (N × H)
226. #541	C₄×C₆₇	semidirect
261. C₄⋊Q₈	C₂×C₄×C₆₇	semidirect
265. #125	C₂×C₄×C₆₇	semidirect
269. C₄×Q₈	C₂×C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#541×C₆₇	2
C₂×#433×C₆₇	2
C₆₇×#2317	2
C₆₇×#1992	2
C₆₇×#1991	2
C₆₇×#1960	2
C₆₇×#1956	2
C₆₇×#1873	2
C₆₇×#1872	2
C₆₇×#1805	2
C₆₇×#1803	2
#21873	67

Subgroups — C67×#21873 #30701124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21873 #30701124