Subgroups — C₂×C₆₇×#2612 #30714121

Order 17152.

2030 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2612	C₂	1	16	yes	maximal, direct product factor
2	C₂×#400×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#398×C₆₇	C₂	1	2	yes	maximal
4	C₂×#395×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#391×C₆₇	C₂	1	2	yes	maximal
6	C₂×#389×C₆₇	C₂	1	2	yes	maximal
7	C₂×#387×C₆₇	C₂	1	2	yes	maximal
8	C₂×#387×C₆₇	C₂	1	2	yes	maximal
9	C₂×#382×C₆₇	C₂	1	1	yes	maximal, characteristic
10	C₂×#376×C₆₇	C₂	1	2	yes	maximal
11	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
13	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
14	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
15	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
19	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
20	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
21	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
22	#400×C₆₇	C₂²	1	8	yes	—
23	#398×C₆₇	C₂²	1	16	yes	—
24	#395×C₆₇	C₂²	1	8	yes	—
25	#391×C₆₇	C₂²	1	16	yes	—
26	#389×C₆₇	C₂²	1	16	yes	—
27	#387×C₆₇	C₂²	1	16	yes	—
28	#387×C₆₇	C₂²	1	16	yes	—
29	#382×C₆₇	C₂²	1	8	yes	—
30	#376×C₆₇	C₂²	1	16	yes	—
31	C₂×#95×C₆₇	C₂²	1	2	yes	—
32	C₂×#95×C₆₇	C₂²	1	2	yes	—
33	C₂×#95×C₆₇	C₂²	1	2	yes	—
34	C₂×#95×C₆₇	C₂²	1	2	yes	—
35	C₂×#95×C₆₇	C₂²	1	2	yes	—
36	C₂×#95×C₆₇	C₂²	1	2	yes	—
37	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#95×C₆₇	C₂²	1	2	yes	—
40	C₂×#95×C₆₇	C₂²	1	2	yes	—
41	C₂²×Q₈×C₆₇	—	2	8	no	—
42	C₂²×Q₈×C₆₇	—	4	2	no	—
43	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
44	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
45	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
46	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
47	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
50	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
52	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
56	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
57	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
58	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
59	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
60	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
61	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
62	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
63	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
66	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
67	C₂×C₄²×C₆₇	—	2	8	no	—
68	C₂×C₄²×C₆₇	—	2	8	no	—
69	C₂×C₄²×C₆₇	Q₈	1	8	yes	complemented
70	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
71	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
72	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
73	#95×C₆₇	C₂³	1	8	yes	—
74	#95×C₆₇	C₂³	1	8	yes	—
75	#95×C₆₇	C₂³	1	8	yes	—
76	#95×C₆₇	C₂³	1	8	yes	—
77	#95×C₆₇	C₂³	1	8	yes	—
78	#95×C₆₇	C₂³	1	8	yes	—
79	#95×C₆₇	C₂³	1	4	yes	—
80	#95×C₆₇	C₂³	1	4	yes	—
81	#95×C₆₇	C₂³	1	8	yes	—
82	#95×C₆₇	C₂³	1	8	yes	—
83	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
84	C₂×Q₈×C₆₇	—	2	16	no	—
85	C₂×Q₈×C₆₇	—	4	4	no	—
86	C₂×Q₈×C₆₇	—	4	16	no	—
87	C₂²×C₄×C₆₇	—	2	4	no	—
88	C₂²×C₄×C₆₇	—	2	4	no	—
89	C₂²×C₄×C₆₇	—	2	4	no	—
90	C₂²×C₄×C₆₇	—	2	4	no	—
91	C₂²×C₄×C₆₇	—	2	2	no	—
92	C₂²×C₄×C₆₇	—	2	4	no	—
93	C₂²×C₄×C₆₇	—	2	2	no	—
94	C₂²×C₄×C₆₇	—	2	4	no	—
95	C₂²×C₄×C₆₇	—	2	4	no	—
96	C₂²×C₄×C₆₇	—	2	4	no	—
97	C₂²×C₄×C₆₇	—	2	4	no	—
98	C₂²×C₄×C₆₇	—	2	4	no	—
99	C₂²×C₄×C₆₇	—	2	4	no	—
100	C₂²×C₄×C₆₇	—	2	4	no	—
101	C₂²×C₄×C₆₇	—	2	4	no	—
102	C₂²×C₄×C₆₇	—	2	4	no	—
103	C₂²×C₄×C₆₇	—	2	2	no	—
104	C₂²×C₄×C₆₇	—	2	4	no	—
105	C₂²×C₄×C₆₇	—	2	4	no	—
106	C₂²×C₄×C₆₇	—	2	2	no	—
107	C₂²×C₄×C₆₇	—	2	4	no	—
108	C₂²×C₄×C₆₇	C₂×Q₈	1	8	yes	—
109	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
110	C₂²×C₄×C₆₇	#41	1	8	yes	—
111	C₂²×C₄×C₆₇	#41	1	8	yes	—
112	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
113	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
114	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
115	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
116	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
117	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
118	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
119	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
120	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
121	C₄⋊C₄×C₆₇	—	2	16	no	—
122	C₄⋊C₄×C₆₇	—	2	16	no	—
123	C₄⋊C₄×C₆₇	—	2	16	no	—
124	C₄⋊C₄×C₆₇	—	4	4	no	—
125	C₄⋊C₄×C₆₇	—	4	4	no	—
126	C₄⋊C₄×C₆₇	—	4	8	no	—
127	C₄⋊C₄×C₆₇	—	4	4	no	—
128	C₄²×C₆₇	—	2	16	no	—
129	C₄²×C₆₇	—	2	16	no	—
130	C₄²×C₆₇	C₂×Q₈	1	16	yes	complemented
131	C₂³×C₆₇	C₂²×Q₈	1	2	yes	—
132	C₂³×C₆₇	C₂×#41	1	4	yes	—
133	C₂³×C₆₇	C₂×#41	1	4	yes	—
134	C₂³×C₆₇	H(5,𝔽₂)	1	2	yes	—
135	C₂³×C₆₇	2₋¹⁺⁴	1	2	yes	—
136	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
137	Q₈×C₆₇	—	4	16	no	—
138	C₂×C₄×C₆₇	—	2	16	no	—
139	C₂×C₄×C₆₇	—	2	8	no	—
140	C₂×C₄×C₆₇	—	2	16	no	—
141	C₂×C₄×C₆₇	—	2	16	no	—
142	C₂×C₄×C₆₇	—	2	4	no	—
143	C₂×C₄×C₆₇	—	2	4	no	—
144	C₂×C₄×C₆₇	—	2	4	no	—
145	C₂×C₄×C₆₇	—	2	4	no	—
146	C₂×C₄×C₆₇	—	2	2	no	—
147	C₂×C₄×C₆₇	—	2	4	no	—
148	C₂×C₄×C₆₇	—	2	2	no	—
149	C₂×C₄×C₆₇	—	2	4	no	—
150	C₂×C₄×C₆₇	—	2	4	no	—
151	C₂×C₄×C₆₇	—	2	4	no	—
152	C₂×C₄×C₆₇	—	2	4	no	—
153	C₂×C₄×C₆₇	—	2	4	no	—
154	C₂×C₄×C₆₇	—	2	4	no	—
155	C₂×C₄×C₆₇	—	2	4	no	—
156	C₂×C₄×C₆₇	—	2	4	no	—
157	C₂×C₄×C₆₇	—	2	4	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	2	no	—
162	C₂×C₄×C₆₇	—	2	4	no	—
163	C₂×C₄×C₆₇	—	4	4	no	—
164	C₂×C₄×C₆₇	—	4	4	no	—
165	C₂×C₄×C₆₇	—	4	8	no	—
166	C₂×C₄×C₆₇	—	4	8	no	—
167	C₂×C₄×C₆₇	—	4	8	no	—
168	C₂×C₄×C₆₇	C₂²×Q₈	1	8	yes	—
169	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
170	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
171	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
172	C₂²×C₆₇	#533	1	8	yes	—
173	C₂²×C₆₇	#544	1	4	yes	—
174	C₂²×C₆₇	#551	1	4	yes	—
175	C₂²×C₆₇	#552	1	8	yes	—
176	C₂²×C₆₇	#555	1	4	yes	—
177	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
178	C₂²×C₆₇	C₂²×#41	1	2	yes	—
179	C₂²×C₆₇	C₂²×#41	1	2	yes	—
180	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
181	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
182	C₄×C₆₇	—	2	8	no	—
183	C₄×C₆₇	—	2	4	no	—
184	C₄×C₆₇	—	2	8	no	—
185	C₄×C₆₇	—	2	8	no	—
186	C₄×C₆₇	—	4	2	no	—
187	C₄×C₆₇	—	4	2	no	—
188	C₄×C₆₇	—	4	4	no	—
189	C₄×C₆₇	—	4	4	no	—
190	C₄×C₆₇	—	4	4	no	—
191	C₂×#2612	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
192	C₂×C₆₇	#2612	1	8	yes	direct product factor
193	C₂×C₆₇	C₂×#533	1	2	yes	—
194	C₂×C₆₇	C₂×#544	1	1	yes	characteristic
195	C₂×C₆₇	C₂×#551	1	1	yes	characteristic
196	C₂×C₆₇	C₂×#552	1	2	yes	—
197	C₂×C₆₇	C₂×#555	1	1	yes	characteristic
198	#2612	C₂×C₆₇	1	16	yes	direct product factor
199	C₂×#400	C₂×C₆₇	1	1	yes	characteristic
200	C₂×#398	C₂×C₆₇	1	2	yes	—
201	C₂×#395	C₂×C₆₇	1	1	yes	characteristic
202	C₂×#391	C₂×C₆₇	1	2	yes	—
203	C₂×#389	C₂×C₆₇	1	2	yes	—
204	C₂×#387	C₂×C₆₇	1	2	yes	—
205	C₂×#387	C₂×C₆₇	1	2	yes	—
206	C₂×#382	C₂×C₆₇	1	1	yes	characteristic
207	C₂×#376	C₂×C₆₇	1	2	yes	—
208	C₆₇	C₂×#2612	1	1	yes	characteristic, direct product factor, Sylow
209	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
210	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
211	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
212	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
213	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
214	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
215	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
216	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
217	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
218	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
219	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
220	#400	C₂²×C₆₇	1	8	yes	—
221	#398	C₂²×C₆₇	1	16	yes	—
222	#395	C₂²×C₆₇	1	8	yes	—
223	#391	C₂²×C₆₇	1	16	yes	—
224	#389	C₂²×C₆₇	1	16	yes	—
225	#387	C₂²×C₆₇	1	16	yes	—
226	#387	C₂²×C₆₇	1	16	yes	—
227	#382	C₂²×C₆₇	1	8	yes	—
228	#376	C₂²×C₆₇	1	16	yes	—
229	C₂×#95	C₂²×C₆₇	1	2	yes	—
230	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
231	C₂×#95	C₂²×C₆₇	1	2	yes	—
232	C₂×#95	C₂²×C₆₇	1	2	yes	—
233	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
234	C₂×#95	C₂²×C₆₇	1	2	yes	—
235	C₂×#95	C₂²×C₆₇	1	2	yes	—
236	C₂×#95	C₂²×C₆₇	1	2	yes	—
237	C₂×#95	C₂²×C₆₇	1	2	yes	—
238	C₂×#95	C₂²×C₆₇	1	2	yes	—
239	C₂²×Q₈	—	2	8	no	—
240	C₂²×Q₈	—	4	2	no	—
241	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
242	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
243	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
244	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
245	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
246	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
247	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
248	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
249	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
250	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
251	C₂×C₄⋊C₄	—	2	8	no	—
252	C₂×C₄⋊C₄	—	2	8	no	—
253	C₂×C₄⋊C₄	—	2	8	no	—
254	C₂×C₄⋊C₄	—	4	2	no	—
255	C₂×C₄⋊C₄	—	4	2	no	—
256	C₂×C₄⋊C₄	—	4	4	no	—
257	C₂×C₄⋊C₄	—	4	2	no	—
258	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
259	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
260	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
261	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
262	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
263	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
264	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
265	C₂×C₄²	—	2	8	no	—
266	C₂×C₄²	—	2	8	no	—
267	C₂×C₄²	Q₈×C₆₇	1	8	yes	complemented
268	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
269	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
270	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
271	#95	C₂³×C₆₇	1	8	yes	—
272	#95	C₂³×C₆₇	1	8	yes	—
273	#95	C₂³×C₆₇	1	8	yes	—
274	#95	C₂³×C₆₇	1	8	yes	—
275	#95	C₂³×C₆₇	1	8	yes	—
276	#95	C₂³×C₆₇	1	8	yes	—
277	#95	C₂³×C₆₇	1	4	yes	—
278	#95	C₂³×C₆₇	1	4	yes	—
279	#95	C₂³×C₆₇	1	8	yes	—
280	#95	C₂³×C₆₇	1	8	yes	—
281	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
282	C₂×Q₈	—	2	16	no	—
283	C₂×Q₈	—	4	4	no	—
284	C₂×Q₈	—	4	16	no	—
285	C₂²×C₄	—	2	4	no	—
286	C₂²×C₄	—	2	4	no	—
287	C₂²×C₄	—	2	4	no	—
288	C₂²×C₄	—	2	4	no	—
289	C₂²×C₄	—	2	2	no	—
290	C₂²×C₄	—	2	4	no	—
291	C₂²×C₄	—	2	2	no	—
292	C₂²×C₄	—	2	4	no	—
293	C₂²×C₄	—	2	4	no	—
294	C₂²×C₄	—	2	4	no	—
295	C₂²×C₄	—	2	4	no	—
296	C₂²×C₄	—	2	4	no	—
297	C₂²×C₄	—	2	4	no	—
298	C₂²×C₄	—	2	4	no	—
299	C₂²×C₄	—	2	4	no	—
300	C₂²×C₄	—	2	4	no	—
301	C₂²×C₄	—	2	2	no	—
302	C₂²×C₄	—	2	4	no	—
303	C₂²×C₄	—	2	4	no	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	—	2	4	no	—
306	C₂²×C₄	C₂×Q₈×C₆₇	1	8	yes	—
307	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
308	C₂²×C₄	#41×C₆₇	1	8	yes	—
309	C₂²×C₄	#41×C₆₇	1	8	yes	—
310	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
311	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
312	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
313	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
314	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
315	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
316	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
317	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
318	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
319	C₄⋊C₄	—	2	16	no	—
320	C₄⋊C₄	—	2	16	no	—
321	C₄⋊C₄	—	2	16	no	—
322	C₄⋊C₄	—	4	4	no	—
323	C₄⋊C₄	—	4	4	no	—
324	C₄⋊C₄	—	4	8	no	—
325	C₄⋊C₄	—	4	4	no	—
326	C₄²	—	2	16	no	—
327	C₄²	—	2	16	no	—
328	C₄²	C₂×Q₈×C₆₇	1	16	yes	complemented
329	C₂³	C₂²×Q₈×C₆₇	1	2	yes	—
330	C₂³	C₂×#41×C₆₇	1	4	yes	—
331	C₂³	C₂×#41×C₆₇	1	4	yes	—
332	C₂³	H(5,𝔽₂)×C₆₇	1	2	yes	—
333	C₂³	2₋¹⁺⁴×C₆₇	1	2	yes	—
334	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
335	Q₈	—	4	16	no	—
336	C₂×C₄	—	2	16	no	—
337	C₂×C₄	—	2	8	no	—
338	C₂×C₄	—	2	16	no	—
339	C₂×C₄	—	2	16	no	—
340	C₂×C₄	—	2	4	no	—
341	C₂×C₄	—	2	4	no	—
342	C₂×C₄	—	2	4	no	—
343	C₂×C₄	—	2	4	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	4	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	4	no	—
348	C₂×C₄	—	2	4	no	—
349	C₂×C₄	—	2	4	no	—
350	C₂×C₄	—	2	4	no	—
351	C₂×C₄	—	2	4	no	—
352	C₂×C₄	—	2	4	no	—
353	C₂×C₄	—	2	4	no	—
354	C₂×C₄	—	2	4	no	—
355	C₂×C₄	—	2	4	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	4	no	—
358	C₂×C₄	—	2	4	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	4	no	—
361	C₂×C₄	—	4	4	no	—
362	C₂×C₄	—	4	4	no	—
363	C₂×C₄	—	4	8	no	—
364	C₂×C₄	—	4	8	no	—
365	C₂×C₄	—	4	8	no	—
366	C₂×C₄	C₂²×Q₈×C₆₇	1	8	yes	—
367	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
368	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
369	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
370	C₂²	#533×C₆₇	1	8	yes	—
371	C₂²	#544×C₆₇	1	4	yes	—
372	C₂²	#551×C₆₇	1	4	yes	—
373	C₂²	#552×C₆₇	1	8	yes	—
374	C₂²	#555×C₆₇	1	4	yes	—
375	C₂²	C₂³×Q₈×C₆₇	1	1	yes	characteristic
376	C₂²	C₂²×#41×C₆₇	1	2	yes	—
377	C₂²	C₂²×#41×C₆₇	1	2	yes	—
378	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
379	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
380	C₄	—	2	8	no	—
381	C₄	—	2	4	no	—
382	C₄	—	2	8	no	—
383	C₄	—	2	8	no	—
384	C₄	—	4	2	no	—
385	C₄	—	4	2	no	—
386	C₄	—	4	4	no	—
387	C₄	—	4	4	no	—
388	C₄	—	4	4	no	—
389	C₂	C₆₇×#2612	1	8	yes	direct product factor
390	C₂	C₂×#533×C₆₇	1	2	yes	—
391	C₂	C₂×#544×C₆₇	1	1	yes	characteristic
392	C₂	C₂×#551×C₆₇	1	1	yes	characteristic
393	C₂	C₂×#552×C₆₇	1	2	yes	—
394	C₂	C₂×#555×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#2612	C₂	direct (N × H)
69. C₂×C₄²×C₆₇	Q₈	semidirect
130. C₄²×C₆₇	C₂×Q₈	semidirect
191. C₂×#2612	C₆₇	direct (N × H)
192. C₂×C₆₇	#2612	direct (N × H)
198. #2612	C₂×C₆₇	direct (N × H)
208. C₆₇	C₂×#2612	direct (N × H)
267. C₂×C₄²	Q₈×C₆₇	semidirect
328. C₄²	C₂×Q₈×C₆₇	semidirect
389. C₂	C₆₇×#2612	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#2612	2
C₂×#400×C₆₇	2
C₂×#398×C₆₇	2
C₂×#395×C₆₇	2
C₂×#391×C₆₇	2
C₂×#389×C₆₇	2
C₂×#387×C₆₇	2
C₂×#382×C₆₇	2
C₂×#376×C₆₇	2
C₂×#2612	67

Subgroups — C2×C67×#2612 #30714121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2612 #30714121