Subgroups — C₂×C₆₇×#2548 #30714184

Order 17152.

3958 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#127×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#124×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂²×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2548	C₂	1	16	yes	maximal, direct product factor
5	C₂×#394×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₂×#390×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#390×C₆₇	C₂	1	4	yes	maximal, complemented
8	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂³×D₄×C₆₇	C₂²	1	2	yes	—
11	C₂³×D₄×C₆₇	C₂²	1	2	yes	—
12	C₂³×D₄×C₆₇	C₂²	1	2	yes	—
13	C₂×#127×C₆₇	—	2	8	no	—
14	C₂×#127×C₆₇	C₂²	1	8	yes	—
15	C₂×#124×C₆₇	—	2	16	no	—
16	C₂×#124×C₆₇	C₂²	1	16	yes	complemented
17	C₂×C₄×Q₈×C₆₇	—	2	8	no	—
18	C₂×C₄×Q₈×C₆₇	C₂²	1	8	yes	complemented
19	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
20	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
22	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	—
23	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	—
24	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
25	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
26	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
27	#394×C₆₇	C₂²	1	32	yes	complemented
28	#390×C₆₇	C₂²	1	32	yes	complemented
29	#390×C₆₇	C₂²	1	16	yes	complemented
30	C₄×C₄⋊C₄×C₆₇	C₂²	1	8	yes	complemented
31	C₂×#95×C₆₇	C₂²	1	2	yes	—
32	C₂⁵×C₆₇	C₂³	1	4	yes	—
33	C₂²×Q₈×C₆₇	—	2	4	no	—
34	C₂²×Q₈×C₆₇	—	2	4	no	—
35	C₂²×Q₈×C₆₇	D₄	1	8	yes	complemented
36	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
37	C₂²×D₄×C₆₇	—	2	8	no	—
38	C₂²×D₄×C₆₇	—	2	8	no	—
39	C₂²×D₄×C₆₇	—	2	8	no	—
40	C₂²×D₄×C₆₇	—	2	8	no	—
41	C₂²×D₄×C₆₇	—	2	8	no	—
42	C₂²×D₄×C₆₇	—	2	16	no	—
43	C₂²×D₄×C₆₇	—	4	8	no	—
44	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
45	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
46	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
47	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
48	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
49	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
50	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
52	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
53	#127×C₆₇	—	2	32	no	—
54	#124×C₆₇	—	2	64	no	—
55	C₄×Q₈×C₆₇	—	2	32	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
59	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
60	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
61	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
62	C₂×C₂²⋊C₄×C₆₇	—	2	32	no	—
63	C₂×C₂²⋊C₄×C₆₇	—	4	16	no	—
64	C₂×C₂²⋊C₄×C₆₇	C₂³	1	32	yes	—
65	C₂×C₂²⋊C₄×C₆₇	C₂³	1	32	yes	—
66	C₂×C₄²×C₆₇	—	2	8	no	—
67	C₂×C₄²×C₆₇	—	2	8	no	—
68	C₂×C₄²×C₆₇	D₄	1	8	yes	—
69	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
70	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
71	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
72	#95×C₆₇	C₂³	1	8	yes	—
73	C₂⁴×C₆₇	—	2	8	no	—
74	C₂⁴×C₆₇	—	2	8	no	—
75	C₂⁴×C₆₇	—	2	8	no	—
76	C₂⁴×C₆₇	—	2	16	no	—
77	C₂⁴×C₆₇	—	2	16	no	—
78	C₂⁴×C₆₇	C₂⁴	1	8	yes	—
79	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
80	C₂×Q₈×C₆₇	—	2	32	no	—
81	C₂×Q₈×C₆₇	—	2	8	no	—
82	C₂×Q₈×C₆₇	—	2	8	no	—
83	C₂×Q₈×C₆₇	C₂×D₄	1	16	yes	complemented
84	C₂×D₄×C₆₇	—	2	16	no	—
85	C₂×D₄×C₆₇	—	2	16	no	—
86	C₂×D₄×C₆₇	—	2	16	no	—
87	C₂×D₄×C₆₇	—	2	16	no	—
88	C₂×D₄×C₆₇	—	2	16	no	—
89	C₂×D₄×C₆₇	—	2	32	no	—
90	C₂×D₄×C₆₇	—	4	16	no	—
91	C₂×D₄×C₆₇	—	4	32	no	—
92	C₂×D₄×C₆₇	—	4	32	no	—
93	C₂×D₄×C₆₇	—	4	32	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	8	no	—
98	C₂²×C₄×C₆₇	—	2	4	no	—
99	C₂²×C₄×C₆₇	—	2	8	no	—
100	C₂²×C₄×C₆₇	—	2	8	no	—
101	C₂²×C₄×C₆₇	—	2	8	no	—
102	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
103	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
104	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
105	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
106	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
107	C₂²×C₄×C₆₇	#41	1	4	yes	—
108	C₂²×C₄×C₆₇	#41	1	8	yes	—
109	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
110	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
111	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
112	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
113	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
114	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
115	C₄⋊C₄×C₆₇	—	2	16	no	—
116	C₄⋊C₄×C₆₇	—	2	8	no	—
117	C₄⋊C₄×C₆₇	—	2	16	no	—
118	C₂²⋊C₄×C₆₇	—	2	64	no	—
119	C₂²⋊C₄×C₆₇	—	4	32	no	—
120	C₄²×C₆₇	—	2	16	no	—
121	C₄²×C₆₇	—	2	16	no	—
122	C₄²×C₆₇	C₂×D₄	1	16	yes	—
123	C₂³×C₆₇	—	2	8	no	—
124	C₂³×C₆₇	—	2	8	no	—
125	C₂³×C₆₇	—	2	8	no	—
126	C₂³×C₆₇	—	2	16	no	—
127	C₂³×C₆₇	—	2	16	no	—
128	C₂³×C₆₇	—	4	16	no	—
129	C₂³×C₆₇	—	4	16	no	—
130	C₂³×C₆₇	—	4	16	no	—
131	C₂³×C₆₇	—	4	32	no	—
132	C₂³×C₆₇	—	4	32	no	—
133	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
134	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
135	C₂³×C₆₇	C₂×#41	1	2	yes	—
136	C₂³×C₆₇	C₂×#41	1	4	yes	—
137	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	—
138	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
139	Q₈×C₆₇	—	2	32	no	—
140	D₄×C₆₇	—	4	32	no	—
141	D₄×C₆₇	—	4	32	no	—
142	D₄×C₆₇	—	4	32	no	—
143	C₂×C₄×C₆₇	—	2	16	no	—
144	C₂×C₄×C₆₇	—	2	8	no	—
145	C₂×C₄×C₆₇	—	2	4	no	—
146	C₂×C₄×C₆₇	—	2	4	no	—
147	C₂×C₄×C₆₇	—	2	4	no	—
148	C₂×C₄×C₆₇	—	2	8	no	—
149	C₂×C₄×C₆₇	—	2	4	no	—
150	C₂×C₄×C₆₇	—	2	8	no	—
151	C₂×C₄×C₆₇	—	2	8	no	—
152	C₂×C₄×C₆₇	—	2	8	no	—
153	C₂×C₄×C₆₇	—	4	8	no	—
154	C₂×C₄×C₆₇	—	4	16	no	—
155	C₂×C₄×C₆₇	#121	1	16	yes	—
156	C₂×C₄×C₆₇	#124	1	16	yes	complemented
157	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
158	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
159	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
160	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
161	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
162	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
163	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
164	C₂²×C₆₇	—	4	8	no	—
165	C₂²×C₆₇	—	4	8	no	—
166	C₂²×C₆₇	—	4	8	no	—
167	C₂²×C₆₇	—	4	16	no	—
168	C₂²×C₆₇	—	4	16	no	—
169	C₂²×C₆₇	—	8	32	no	—
170	C₂²×C₆₇	C₂×#121	1	4	yes	—
171	C₂²×C₆₇	C₂×#124	1	4	yes	—
172	C₂²×C₆₇	#535	1	4	yes	—
173	C₂²×C₆₇	#550	1	8	yes	—
174	C₂²×C₆₇	#559	1	8	yes	—
175	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
176	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
177	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
178	C₂²×C₆₇	C₂²×#41	1	2	yes	—
179	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
180	C₄×C₆₇	—	2	8	no	—
181	C₄×C₆₇	—	2	4	no	—
182	C₄×C₆₇	—	4	4	no	—
183	C₄×C₆₇	—	4	8	no	—
184	C₄×C₆₇	C₂×#121	1	8	yes	—
185	C₄×C₆₇	C₂×#124	1	8	yes	complemented
186	C₂×#2548	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
187	C₂×C₆₇	—	8	8	no	—
188	C₂×C₆₇	#2548	1	8	yes	direct product factor
189	C₂×C₆₇	C₂²×#121	1	1	yes	characteristic
190	C₂×C₆₇	C₂²×#124	1	1	yes	characteristic
191	C₂×C₆₇	C₂×#535	1	1	yes	characteristic
192	C₂×C₆₇	C₂×#550	1	2	yes	—
193	C₂×C₆₇	C₂×#559	1	2	yes	—
194	C₂²×#127	C₂×C₆₇	1	1	yes	characteristic
195	C₂²×#124	C₂×C₆₇	1	2	yes	complemented
196	C₂²×C₄×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
197	#2548	C₂×C₆₇	1	16	yes	direct product factor
198	C₂×#394	C₂×C₆₇	1	4	yes	complemented
199	C₂×#390	C₂×C₆₇	1	4	yes	complemented
200	C₂×#390	C₂×C₆₇	1	2	yes	complemented
201	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
202	C₆₇	C₂×#2548	1	1	yes	characteristic, direct product factor, Sylow
203	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
204	C₂³×D₄	C₂²×C₆₇	1	2	yes	—
205	C₂³×D₄	C₂²×C₆₇	1	2	yes	—
206	C₂³×D₄	C₂²×C₆₇	1	2	yes	—
207	C₂×#127	—	2	8	no	—
208	C₂×#127	C₂²×C₆₇	1	8	yes	—
209	C₂×#124	—	2	16	no	—
210	C₂×#124	C₂²×C₆₇	1	16	yes	complemented
211	C₂×C₄×Q₈	—	2	8	no	—
212	C₂×C₄×Q₈	C₂²×C₆₇	1	8	yes	complemented
213	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
214	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
215	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
216	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	—
217	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	—
218	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
219	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
220	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
221	#394	C₂²×C₆₇	1	32	yes	complemented
222	#390	C₂²×C₆₇	1	32	yes	complemented
223	#390	C₂²×C₆₇	1	16	yes	complemented
224	C₄×C₄⋊C₄	C₂²×C₆₇	1	8	yes	complemented
225	C₂×#95	C₂²×C₆₇	1	2	yes	—
226	C₂⁵	C₂³×C₆₇	1	4	yes	—
227	C₂²×Q₈	—	2	4	no	—
228	C₂²×Q₈	—	2	4	no	—
229	C₂²×Q₈	D₄×C₆₇	1	8	yes	complemented
230	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
231	C₂²×D₄	—	2	8	no	—
232	C₂²×D₄	—	2	8	no	—
233	C₂²×D₄	—	2	8	no	—
234	C₂²×D₄	—	2	8	no	—
235	C₂²×D₄	—	2	8	no	—
236	C₂²×D₄	—	2	16	no	—
237	C₂²×D₄	—	4	8	no	—
238	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
239	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
240	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
241	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
242	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
243	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
244	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
245	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
246	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
247	#127	—	2	32	no	—
248	#124	—	2	64	no	—
249	C₄×Q₈	—	2	32	no	—
250	C₂×C₄⋊C₄	—	2	8	no	—
251	C₂×C₄⋊C₄	—	2	4	no	—
252	C₂×C₄⋊C₄	—	2	8	no	—
253	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
254	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
255	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
256	C₂×C₂²⋊C₄	—	2	32	no	—
257	C₂×C₂²⋊C₄	—	4	16	no	—
258	C₂×C₂²⋊C₄	C₂³×C₆₇	1	32	yes	—
259	C₂×C₂²⋊C₄	C₂³×C₆₇	1	32	yes	—
260	C₂×C₄²	—	2	8	no	—
261	C₂×C₄²	—	2	8	no	—
262	C₂×C₄²	D₄×C₆₇	1	8	yes	—
263	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
264	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
265	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
266	#95	C₂³×C₆₇	1	8	yes	—
267	C₂⁴	—	2	8	no	—
268	C₂⁴	—	2	8	no	—
269	C₂⁴	—	2	8	no	—
270	C₂⁴	—	2	16	no	—
271	C₂⁴	—	2	16	no	—
272	C₂⁴	C₂⁴×C₆₇	1	8	yes	—
273	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
274	C₂×Q₈	—	2	32	no	—
275	C₂×Q₈	—	2	8	no	—
276	C₂×Q₈	—	2	8	no	—
277	C₂×Q₈	C₂×D₄×C₆₇	1	16	yes	complemented
278	C₂×D₄	—	2	16	no	—
279	C₂×D₄	—	2	16	no	—
280	C₂×D₄	—	2	16	no	—
281	C₂×D₄	—	2	16	no	—
282	C₂×D₄	—	2	16	no	—
283	C₂×D₄	—	2	32	no	—
284	C₂×D₄	—	4	16	no	—
285	C₂×D₄	—	4	32	no	—
286	C₂×D₄	—	4	32	no	—
287	C₂×D₄	—	4	32	no	—
288	C₂²×C₄	—	2	4	no	—
289	C₂²×C₄	—	2	4	no	—
290	C₂²×C₄	—	2	4	no	—
291	C₂²×C₄	—	2	8	no	—
292	C₂²×C₄	—	2	4	no	—
293	C₂²×C₄	—	2	8	no	—
294	C₂²×C₄	—	2	8	no	—
295	C₂²×C₄	—	2	8	no	—
296	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
297	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
298	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
299	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
300	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
301	C₂²×C₄	#41×C₆₇	1	4	yes	—
302	C₂²×C₄	#41×C₆₇	1	8	yes	—
303	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
304	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
305	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
306	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
307	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
308	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
309	C₄⋊C₄	—	2	16	no	—
310	C₄⋊C₄	—	2	8	no	—
311	C₄⋊C₄	—	2	16	no	—
312	C₂²⋊C₄	—	2	64	no	—
313	C₂²⋊C₄	—	4	32	no	—
314	C₄²	—	2	16	no	—
315	C₄²	—	2	16	no	—
316	C₄²	C₂×D₄×C₆₇	1	16	yes	—
317	C₂³	—	2	8	no	—
318	C₂³	—	2	8	no	—
319	C₂³	—	2	8	no	—
320	C₂³	—	2	16	no	—
321	C₂³	—	2	16	no	—
322	C₂³	—	4	16	no	—
323	C₂³	—	4	16	no	—
324	C₂³	—	4	16	no	—
325	C₂³	—	4	32	no	—
326	C₂³	—	4	32	no	—
327	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
328	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
329	C₂³	C₂×#41×C₆₇	1	2	yes	—
330	C₂³	C₂×#41×C₆₇	1	4	yes	—
331	C₂³	H(5,𝔽₂)×C₆₇	1	4	yes	—
332	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
333	Q₈	—	2	32	no	—
334	D₄	—	4	32	no	—
335	D₄	—	4	32	no	—
336	D₄	—	4	32	no	—
337	C₂×C₄	—	2	16	no	—
338	C₂×C₄	—	2	8	no	—
339	C₂×C₄	—	2	4	no	—
340	C₂×C₄	—	2	4	no	—
341	C₂×C₄	—	2	4	no	—
342	C₂×C₄	—	2	8	no	—
343	C₂×C₄	—	2	4	no	—
344	C₂×C₄	—	2	8	no	—
345	C₂×C₄	—	2	8	no	—
346	C₂×C₄	—	2	8	no	—
347	C₂×C₄	—	4	8	no	—
348	C₂×C₄	—	4	16	no	—
349	C₂×C₄	#121×C₆₇	1	16	yes	—
350	C₂×C₄	#124×C₆₇	1	16	yes	complemented
351	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
352	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
353	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
354	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
355	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
356	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
357	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
358	C₂²	—	4	8	no	—
359	C₂²	—	4	8	no	—
360	C₂²	—	4	8	no	—
361	C₂²	—	4	16	no	—
362	C₂²	—	4	16	no	—
363	C₂²	—	8	32	no	—
364	C₂²	C₂×#121×C₆₇	1	4	yes	—
365	C₂²	C₂×#124×C₆₇	1	4	yes	—
366	C₂²	#535×C₆₇	1	4	yes	—
367	C₂²	#550×C₆₇	1	8	yes	—
368	C₂²	#559×C₆₇	1	8	yes	—
369	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
370	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
371	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
372	C₂²	C₂²×#41×C₆₇	1	2	yes	—
373	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
374	C₄	—	2	8	no	—
375	C₄	—	2	4	no	—
376	C₄	—	4	4	no	—
377	C₄	—	4	8	no	—
378	C₄	C₂×#121×C₆₇	1	8	yes	—
379	C₄	C₂×#124×C₆₇	1	8	yes	complemented
380	C₂	—	8	8	no	—
381	C₂	C₆₇×#2548	1	8	yes	direct product factor
382	C₂	C₂²×#121×C₆₇	1	1	yes	characteristic
383	C₂	C₂²×#124×C₆₇	1	1	yes	characteristic
384	C₂	C₂×#535×C₆₇	1	1	yes	characteristic
385	C₂	C₂×#550×C₆₇	1	2	yes	—
386	C₂	C₂×#559×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂²×#127×C₆₇	C₂
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
2. C₂²×#124×C₆₇	C₂	semidirect
3. C₂²×C₄×Q₈×C₆₇	C₂	semidirect
4. C₆₇×#2548	C₂	direct (N × H)
5. C₂×#394×C₆₇	C₂	semidirect
6. C₂×#390×C₆₇	C₂	semidirect
7. C₂×#390×C₆₇	C₂	semidirect
8. C₂×C₄×C₄⋊C₄×C₆₇	C₂	semidirect
16. C₂×#124×C₆₇	C₂²	semidirect
18. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
27. #394×C₆₇	C₂²	semidirect
28. #390×C₆₇	C₂²	semidirect
29. #390×C₆₇	C₂²	semidirect
30. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
35. C₂²×Q₈×C₆₇	D₄	semidirect
83. C₂×Q₈×C₆₇	C₂×D₄	semidirect
156. C₂×C₄×C₆₇	#124	semidirect
185. C₄×C₆₇	C₂×#124	semidirect
186. C₂×#2548	C₆₇	direct (N × H)
188. C₂×C₆₇	#2548	direct (N × H)
195. C₂²×#124	C₂×C₆₇	semidirect
196. C₂²×C₄×Q₈	C₂×C₆₇	semidirect
197. #2548	C₂×C₆₇	direct (N × H)
198. C₂×#394	C₂×C₆₇	semidirect
199. C₂×#390	C₂×C₆₇	semidirect
200. C₂×#390	C₂×C₆₇	semidirect
201. C₂×C₄×C₄⋊C₄	C₂×C₆₇	semidirect
202. C₆₇	C₂×#2548	direct (N × H)
210. C₂×#124	C₂²×C₆₇	semidirect
212. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
221. #394	C₂²×C₆₇	semidirect
222. #390	C₂²×C₆₇	semidirect
223. #390	C₂²×C₆₇	semidirect
224. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
229. C₂²×Q₈	D₄×C₆₇	semidirect
277. C₂×Q₈	C₂×D₄×C₆₇	semidirect
350. C₂×C₄	#124×C₆₇	semidirect
379. C₄	C₂×#124×C₆₇	semidirect
381. C₂	C₆₇×#2548	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#127×C₆₇	2
C₂²×#124×C₆₇	2
C₂²×C₄×Q₈×C₆₇	2
C₆₇×#2548	2
C₂×#394×C₆₇	2
C₂×#390×C₆₇	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₂×#2548	67

Subgroups — C2×C67×#2548 #30714184

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2548 #30714184