Subgroups — C₆₇×#21898 #30701149

Order 17152.

1014 conjugacy classes of subgroups.

The subgroup lattice has 412 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×#424×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2314	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1989	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1981	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1894	C₂	1	2	yes	maximal
7	C₆₇×#1892	C₂	1	2	yes	maximal
8	C₆₇×#1805	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1803	C₂	1	2	yes	maximal, complemented
10	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
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	#424×C₆₇	—	2	4	no	—
22	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
23	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
24	C₂×#103×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	#388×C₆₇	—	2	2	no	—
29	#387×C₆₇	—	2	2	no	—
30	#384×C₆₇	C₂²	1	1	yes	characteristic
31	#382×C₆₇	C₂²	1	1	yes	characteristic
32	#380×C₆₇	C₂²	1	2	yes	—
33	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
34	#340×C₆₇	C₂²	1	2	yes	—
35	#340×C₆₇	C₂²	1	2	yes	—
36	#340×C₆₇	C₂²	1	2	yes	—
37	#340×C₆₇	C₂²	1	2	yes	—
38	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
39	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
40	C₂×M₁₆×C₆₇	—	2	2	no	—
41	C₂×M₁₆×C₆₇	—	4	1	no	—
42	C₂×M₁₆×C₆₇	—	4	1	no	—
43	C₂²×C₈×C₆₇	—	2	2	no	—
44	C₂²×C₈×C₆₇	—	4	1	no	—
45	C₂²×C₈×C₆₇	—	4	1	no	—
46	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
47	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
48	C₄⋊Q₈×C₆₇	—	2	2	no	—
49	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
50	#126×C₆₇	—	2	16	no	—
51	#125×C₆₇	—	2	16	no	—
52	#125×C₆₇	—	2	2	no	—
53	#125×C₆₇	C₂×C₄	1	4	yes	—
54	C₂²⋊Q₈×C₆₇	—	2	4	no	—
55	C₂²⋊Q₈×C₆₇	C₂×C₄	1	8	yes	—
56	C₄×Q₈×C₆₇	—	2	4	no	—
57	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	—
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	2	no	—
65	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
66	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
68	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
69	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
70	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
71	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
72	C₂×C₄²×C₆₇	—	2	2	no	—
73	C₂×C₄²×C₆₇	—	2	2	no	—
74	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₆₇	—	2	4	no	—
77	C₄⋊C₈×C₆₇	—	2	4	no	—
78	#103×C₆₇	—	4	2	no	—
79	#103×C₆₇	—	4	2	no	—
80	#103×C₆₇	—	4	2	no	—
81	#103×C₆₇	—	4	2	no	—
82	#95×C₆₇	—	2	2	no	—
83	#95×C₆₇	—	2	2	no	—
84	#95×C₆₇	—	2	2	no	—
85	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
86	C₂×Q₈×C₆₇	—	2	2	no	—
87	C₂×Q₈×C₆₇	—	2	2	no	—
88	C₂×Q₈×C₆₇	—	2	2	no	—
89	C₂×Q₈×C₆₇	—	2	2	no	—
90	C₂×Q₈×C₆₇	—	2	2	no	—
91	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
92	C₂²×C₄×C₆₇	—	2	1	no	—
93	C₂²×C₄×C₆₇	—	2	1	no	—
94	C₂²×C₄×C₆₇	—	2	1	no	—
95	C₂²×C₄×C₆₇	—	2	1	no	—
96	C₂²×C₄×C₆₇	—	2	2	no	—
97	C₂²×C₄×C₆₇	—	2	2	no	—
98	C₂²×C₄×C₆₇	—	2	2	no	—
99	C₂²×C₄×C₆₇	—	2	2	no	—
100	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
106	M₁₆×C₆₇	—	4	4	no	—
107	C₂×C₈×C₆₇	—	2	2	no	—
108	C₂×C₈×C₆₇	—	2	2	no	—
109	C₂×C₈×C₆₇	—	2	2	no	—
110	C₂×C₈×C₆₇	—	2	2	no	—
111	C₂×C₈×C₆₇	—	2	2	no	—
112	C₂×C₈×C₆₇	—	2	2	no	—
113	C₂×C₈×C₆₇	—	4	4	no	—
114	C₄⋊C₄×C₆₇	—	2	8	no	—
115	C₄⋊C₄×C₆₇	—	2	8	no	—
116	C₄⋊C₄×C₆₇	—	2	8	no	—
117	C₄⋊C₄×C₆₇	—	2	8	no	—
118	C₄⋊C₄×C₆₇	—	2	2	no	—
119	C₄⋊C₄×C₆₇	—	2	2	no	—
120	C₄⋊C₄×C₆₇	—	2	2	no	—
121	C₄⋊C₄×C₆₇	—	2	2	no	—
122	C₄⋊C₄×C₆₇	—	2	2	no	—
123	C₄⋊C₄×C₆₇	—	8	2	no	—
124	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
125	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
126	C₂²⋊C₄×C₆₇	—	2	8	no	—
127	C₂²⋊C₄×C₆₇	—	8	2	no	—
128	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
129	C₄²×C₆₇	—	2	8	no	—
130	C₄²×C₆₇	—	8	2	no	—
131	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
132	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
133	C₄²×C₆₇	C₂×D₄	1	2	yes	—
134	C₄²×C₆₇	C₂×D₄	1	2	yes	—
135	C₂³×C₆₇	—	2	1	no	—
136	C₂³×C₆₇	—	2	1	no	—
137	C₂³×C₆₇	—	2	2	no	—
138	C₂³×C₆₇	—	2	2	no	—
139	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
140	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
141	Q₈×C₆₇	—	2	4	no	—
142	Q₈×C₆₇	—	2	4	no	—
143	Q₈×C₆₇	—	4	2	no	—
144	Q₈×C₆₇	—	4	2	no	—
145	C₂×C₄×C₆₇	—	2	1	no	—
146	C₂×C₄×C₆₇	—	2	1	no	—
147	C₂×C₄×C₆₇	—	2	2	no	—
148	C₂×C₄×C₆₇	—	2	2	no	—
149	C₂×C₄×C₆₇	—	2	2	no	—
150	C₂×C₄×C₆₇	—	2	2	no	—
151	C₂×C₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	—	2	2	no	—
153	C₂×C₄×C₆₇	—	2	2	no	—
154	C₂×C₄×C₆₇	—	2	2	no	—
155	C₂×C₄×C₆₇	—	2	4	no	—
156	C₂×C₄×C₆₇	—	2	4	no	—
157	C₂×C₄×C₆₇	—	4	2	no	—
158	C₂×C₄×C₆₇	—	4	2	no	—
159	C₂×C₄×C₆₇	—	4	2	no	—
160	C₂×C₄×C₆₇	—	4	2	no	—
161	C₂×C₄×C₆₇	—	4	2	no	—
162	C₂×C₄×C₆₇	—	4	2	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	C₂×C₂²⋊C₄	1	4	yes	—
168	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
169	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
170	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
171	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
173	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
174	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
175	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
176	C₈×C₆₇	—	4	2	no	—
177	C₈×C₆₇	—	4	2	no	—
178	C₂²×C₆₇	—	2	2	no	—
179	C₂²×C₆₇	—	2	4	no	—
180	C₂²×C₆₇	—	4	2	no	—
181	C₂²×C₆₇	—	4	1	no	—
182	C₂²×C₆₇	—	4	1	no	—
183	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
184	C₂²×C₆₇	#537	1	1	yes	characteristic
185	C₂²×C₆₇	#537	1	1	yes	characteristic
186	C₂²×C₆₇	#577	1	2	yes	—
187	C₂²×C₆₇	#578	1	2	yes	—
188	C₄×C₆₇	—	2	2	no	—
189	C₄×C₆₇	—	2	4	no	—
190	C₄×C₆₇	—	2	4	no	—
191	C₄×C₆₇	—	4	4	no	—
192	C₄×C₆₇	—	4	4	no	—
193	C₄×C₆₇	—	8	2	no	—
194	C₄×C₆₇	—	8	2	no	—
195	C₄×C₆₇	#520	1	1	yes	characteristic
196	C₄×C₆₇	#520	1	1	yes	characteristic
197	C₄×C₆₇	#535	1	1	yes	characteristic
198	C₄×C₆₇	#536	1	1	yes	characteristic
199	#21898	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
200	C₂×C₆₇	—	4	2	no	—
201	C₂×C₆₇	#2311	1	1	yes	characteristic
202	C₂×C₆₇	#2898	1	2	yes	—
203	C₂×C₆₇	#3258	1	2	yes	—
204	C₂×C₆₇	#3260	1	1	yes	characteristic
205	C₂×C₆₇	#3261	1	1	yes	characteristic
206	C₂×#541	C₂×C₆₇	1	1	yes	characteristic
207	C₂×#424	C₂×C₆₇	1	1	yes	characteristic
208	#2314	C₂×C₆₇	1	1	yes	characteristic
209	#1989	C₂×C₆₇	1	2	yes	complemented
210	#1981	C₂×C₆₇	1	2	yes	complemented
211	#1894	C₂×C₆₇	1	2	yes	—
212	#1892	C₂×C₆₇	1	2	yes	—
213	#1805	C₂×C₆₇	1	2	yes	complemented
214	#1803	C₂×C₆₇	1	2	yes	complemented
215	C₆₇	#21898	1	1	yes	characteristic, direct product factor, Sylow
216	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
217	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
218	#541	—	2	4	no	—
219	#541	C₄×C₆₇	1	8	yes	complemented
220	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
221	C₂×#126	—	2	2	no	—
222	C₂×#125	—	2	2	no	—
223	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
224	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
225	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
226	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
227	#424	—	2	4	no	—
228	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
229	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
230	C₂×#103	C₂²×C₆₇	1	2	yes	—
231	C₂×#103	C₂²×C₆₇	1	2	yes	—
232	C₂×#103	C₂²×C₆₇	1	2	yes	—
233	C₂×#103	C₂²×C₆₇	1	2	yes	—
234	#388	—	2	2	no	—
235	#387	—	2	2	no	—
236	#384	C₂²×C₆₇	1	1	yes	characteristic
237	#382	C₂²×C₆₇	1	1	yes	characteristic
238	#380	C₂²×C₆₇	1	2	yes	—
239	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
240	#340	C₂²×C₆₇	1	2	yes	—
241	#340	C₂²×C₆₇	1	2	yes	—
242	#340	C₂²×C₆₇	1	2	yes	—
243	#340	C₂²×C₆₇	1	2	yes	—
244	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
245	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
246	C₂×M₁₆	—	2	2	no	—
247	C₂×M₁₆	—	4	1	no	—
248	C₂×M₁₆	—	4	1	no	—
249	C₂²×C₈	—	2	2	no	—
250	C₂²×C₈	—	4	1	no	—
251	C₂²×C₈	—	4	1	no	—
252	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
253	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
254	C₄⋊Q₈	—	2	2	no	—
255	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
256	#126	—	2	16	no	—
257	#125	—	2	16	no	—
258	#125	—	2	2	no	—
259	#125	C₂×C₄×C₆₇	1	4	yes	—
260	C₂²⋊Q₈	—	2	4	no	—
261	C₂²⋊Q₈	C₂×C₄×C₆₇	1	8	yes	—
262	C₄×Q₈	—	2	4	no	—
263	C₄×Q₈	C₂×C₄×C₆₇	1	8	yes	—
264	#117	C₂×C₄×C₆₇	1	4	yes	—
265	#117	D₄×C₆₇	1	4	yes	—
266	C₂×C₄⋊C₄	—	2	2	no	—
267	C₂×C₄⋊C₄	—	2	2	no	—
268	C₂×C₄⋊C₄	—	2	2	no	—
269	C₂×C₄⋊C₄	—	2	2	no	—
270	C₂×C₄⋊C₄	—	2	2	no	—
271	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
272	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
273	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
274	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
275	C₂×C₂²⋊C₄	—	2	2	no	—
276	C₂×C₂²⋊C₄	—	2	2	no	—
277	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
278	C₂×C₄²	—	2	2	no	—
279	C₂×C₄²	—	2	2	no	—
280	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
281	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
282	C₄⋊C₈	—	2	4	no	—
283	C₄⋊C₈	—	2	4	no	—
284	#103	—	4	2	no	—
285	#103	—	4	2	no	—
286	#103	—	4	2	no	—
287	#103	—	4	2	no	—
288	#95	—	2	2	no	—
289	#95	—	2	2	no	—
290	#95	—	2	2	no	—
291	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
292	C₂×Q₈	—	2	2	no	—
293	C₂×Q₈	—	2	2	no	—
294	C₂×Q₈	—	2	2	no	—
295	C₂×Q₈	—	2	2	no	—
296	C₂×Q₈	—	2	2	no	—
297	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
298	C₂²×C₄	—	2	1	no	—
299	C₂²×C₄	—	2	1	no	—
300	C₂²×C₄	—	2	1	no	—
301	C₂²×C₄	—	2	1	no	—
302	C₂²×C₄	—	2	2	no	—
303	C₂²×C₄	—	2	2	no	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	—	2	2	no	—
306	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
307	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
308	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
309	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
310	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
311	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
312	M₁₆	—	4	4	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	4	4	no	—
320	C₄⋊C₄	—	2	8	no	—
321	C₄⋊C₄	—	2	8	no	—
322	C₄⋊C₄	—	2	8	no	—
323	C₄⋊C₄	—	2	8	no	—
324	C₄⋊C₄	—	2	2	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	2	2	no	—
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	8	2	no	—
330	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
331	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
332	C₂²⋊C₄	—	2	8	no	—
333	C₂²⋊C₄	—	8	2	no	—
334	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
335	C₄²	—	2	8	no	—
336	C₄²	—	8	2	no	—
337	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
338	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
339	C₄²	C₂×D₄×C₆₇	1	2	yes	—
340	C₄²	C₂×D₄×C₆₇	1	2	yes	—
341	C₂³	—	2	1	no	—
342	C₂³	—	2	1	no	—
343	C₂³	—	2	2	no	—
344	C₂³	—	2	2	no	—
345	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
346	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
347	Q₈	—	2	4	no	—
348	Q₈	—	2	4	no	—
349	Q₈	—	4	2	no	—
350	Q₈	—	4	2	no	—
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	4	no	—
362	C₂×C₄	—	2	4	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	2	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	4	2	no	—
371	C₂×C₄	—	4	2	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
374	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
375	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
376	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
377	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
378	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
379	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
380	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
381	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
382	C₈	—	4	2	no	—
383	C₈	—	4	2	no	—
384	C₂²	—	2	2	no	—
385	C₂²	—	2	4	no	—
386	C₂²	—	4	2	no	—
387	C₂²	—	4	1	no	—
388	C₂²	—	4	1	no	—
389	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
390	C₂²	#537×C₆₇	1	1	yes	characteristic
391	C₂²	#537×C₆₇	1	1	yes	characteristic
392	C₂²	#577×C₆₇	1	2	yes	—
393	C₂²	#578×C₆₇	1	2	yes	—
394	C₄	—	2	2	no	—
395	C₄	—	2	4	no	—
396	C₄	—	2	4	no	—
397	C₄	—	4	4	no	—
398	C₄	—	4	4	no	—
399	C₄	—	8	2	no	—
400	C₄	—	8	2	no	—
401	C₄	#520×C₆₇	1	1	yes	characteristic
402	C₄	#520×C₆₇	1	1	yes	characteristic
403	C₄	#535×C₆₇	1	1	yes	characteristic
404	C₄	#536×C₆₇	1	1	yes	characteristic
405	C₂	—	4	2	no	—
406	C₂	C₆₇×#2311	1	1	yes	characteristic
407	C₂	C₆₇×#2898	1	2	yes	—
408	C₂	C₆₇×#3258	1	2	yes	—
409	C₂	C₆₇×#3260	1	1	yes	characteristic
410	C₂	C₆₇×#3261	1	1	yes	characteristic

	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×#1989	C₂	semidirect
5. C₆₇×#1981	C₂	semidirect
8. C₆₇×#1805	C₂	semidirect
9. C₆₇×#1803	C₂	semidirect
13. #541×C₆₇	C₄	semidirect
199. #21898	C₆₇	direct (N × H)
209. #1989	C₂×C₆₇	semidirect
210. #1981	C₂×C₆₇	semidirect
213. #1805	C₂×C₆₇	semidirect
214. #1803	C₂×C₆₇	semidirect
215. C₆₇	#21898	direct (N × H)
219. #541	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×#424×C₆₇	2
C₆₇×#2314	2
C₆₇×#1989	2
C₆₇×#1981	2
C₆₇×#1894	2
C₆₇×#1892	2
C₆₇×#1805	2
C₆₇×#1803	2
#21898	67

Subgroups — C67×#21898 #30701149

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21898 #30701149