Subgroups — C₂×C₆₇×#2883 #30739319

Order 17152.

5338 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×H(5,𝔽₂)×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#518×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2883	C₂	1	16	yes	maximal, direct product factor
4	C₂×#410×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₂²×#99×C₆₇	C₂	1	8	yes	maximal, complemented
6	C₂×H(5,𝔽₂)×C₆₇	—	2	8	no	—
7	C₂×H(5,𝔽₂)×C₆₇	—	2	8	no	—
8	C₂×H(5,𝔽₂)×C₆₇	C₄	1	8	yes	complemented
9	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	8	yes	—
10	C₂²×#41×C₆₇	C₄	1	4	yes	complemented
11	C₂²×#41×C₆₇	C₂²	1	2	yes	—
12	C₂³×D₄×C₆₇	C₄	1	4	yes	complemented
13	C₂³×D₄×C₆₇	C₂²	1	4	yes	—
14	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
15	#518×C₆₇	—	2	8	no	—
16	#518×C₆₇	C₂²	1	16	yes	complemented
17	C₂×C₄×D₄×C₆₇	—	2	8	no	—
18	C₂×#117×C₆₇	C₂²	1	4	yes	complemented
19	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	complemented
20	#410×C₆₇	C₂²	1	32	yes	complemented
21	C₂×#99×C₆₇	C₂²	1	16	yes	complemented
22	C₂×#99×C₆₇	C₂²	1	64	yes	complemented
23	C₂⁵×C₆₇	C₂×C₄	1	4	yes	complemented
24	C₂⁵×C₆₇	C₂³	1	2	yes	—
25	H(5,𝔽₂)×C₆₇	—	2	64	no	—
26	H(5,𝔽₂)×C₆₇	—	2	64	no	—
27	C₂×#41×C₆₇	—	2	16	no	—
28	C₂×#41×C₆₇	—	2	16	no	—
29	C₂×#41×C₆₇	—	2	8	no	—
30	C₂×#41×C₆₇	—	2	8	no	—
31	C₂×#41×C₆₇	C₂×C₄	1	8	yes	complemented
32	C₂×#41×C₆₇	C₂×C₄	1	32	yes	complemented
33	C₂×#41×C₆₇	C₂³	1	8	yes	—
34	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	complemented
35	C₂²×D₄×C₆₇	—	2	16	no	—
36	C₂²×D₄×C₆₇	—	2	16	no	—
37	C₂²×D₄×C₆₇	—	2	16	no	—
38	C₂²×D₄×C₆₇	—	2	16	no	—
39	C₂²×D₄×C₆₇	C₂×C₄	1	4	yes	—
40	C₂²×D₄×C₆₇	C₂×C₄	1	4	yes	—
41	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	complemented
42	C₂²×D₄×C₆₇	C₂×C₄	1	8	yes	complemented
43	C₂²×D₄×C₆₇	C₂×C₄	1	16	yes	complemented
44	C₂²×D₄×C₆₇	C₂×C₄	1	32	yes	complemented
45	C₂²×D₄×C₆₇	D₄	1	8	yes	—
46	C₂²×D₄×C₆₇	D₄	1	8	yes	—
47	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
48	C₂²×D₄×C₆₇	C₂³	1	16	yes	—
49	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
50	C₂³×C₄×C₆₇	—	2	8	no	—
51	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	complemented
52	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	complemented
53	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
54	C₄×D₄×C₆₇	—	2	64	no	—
55	#117×C₆₇	—	2	8	no	—
56	#117×C₆₇	C₂³	1	16	yes	complemented
57	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
58	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
59	C₂×C₂²⋊C₄×C₆₇	—	2	16	no	—
60	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
61	C₂×C₂²⋊C₄×C₆₇	C₂³	1	32	yes	complemented
62	C₂×C₄²×C₆₇	—	2	4	no	—
63	#99×C₆₇	C₂³	1	64	yes	complemented
64	C₂⁴×C₆₇	—	2	8	no	—
65	C₂⁴×C₆₇	—	2	8	no	—
66	C₂⁴×C₆₇	—	8	4	no	—
67	C₂⁴×C₆₇	—	8	8	no	—
68	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
69	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
70	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
71	C₂⁴×C₆₇	C₂²×C₄	1	16	yes	complemented
72	C₂⁴×C₆₇	C₂×D₄	1	8	yes	—
73	C₂⁴×C₆₇	C₂×D₄	1	8	yes	—
74	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
75	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
76	#41×C₆₇	—	2	128	no	—
77	#41×C₆₇	—	2	32	no	—
78	#41×C₆₇	—	2	32	no	—
79	C₂×Q₈×C₆₇	—	2	4	no	—
80	C₂×Q₈×C₆₇	—	2	4	no	—
81	C₂×Q₈×C₆₇	C₂²×C₄	1	8	yes	complemented
82	C₂×D₄×C₆₇	—	2	16	no	—
83	C₂×D₄×C₆₇	—	2	16	no	—
84	C₂×D₄×C₆₇	—	2	64	no	—
85	C₂×D₄×C₆₇	—	2	8	no	—
86	C₂×D₄×C₆₇	—	2	8	no	—
87	C₂×D₄×C₆₇	—	2	64	no	—
88	C₂×D₄×C₆₇	—	2	128	no	—
89	C₂×D₄×C₆₇	—	2	4	no	—
90	C₂×D₄×C₆₇	—	2	4	no	—
91	C₂×D₄×C₆₇	C₂²⋊C₄	1	16	yes	—
92	C₂×D₄×C₆₇	C₂²⋊C₄	1	16	yes	—
93	C₂×D₄×C₆₇	C₂²⋊C₄	1	32	yes	—
94	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
95	C₂×D₄×C₆₇	C₂²×C₄	1	16	yes	—
96	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	complemented
97	C₂×D₄×C₆₇	C₂²×C₄	1	32	yes	complemented
98	C₂×D₄×C₆₇	C₂×D₄	1	32	yes	—
99	C₂×D₄×C₆₇	C₂⁴	1	8	yes	—
100	C₂²×C₄×C₆₇	—	2	8	no	—
101	C₂²×C₄×C₆₇	—	2	8	no	—
102	C₂²×C₄×C₆₇	—	2	8	no	—
103	C₂²×C₄×C₆₇	—	2	8	no	—
104	C₂²×C₄×C₆₇	—	2	8	no	—
105	C₂²×C₄×C₆₇	—	2	32	no	—
106	C₂²×C₄×C₆₇	—	8	16	no	—
107	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
108	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
109	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
110	C₂²×C₄×C₆₇	C₂²×C₄	1	16	yes	complemented
111	C₂²×C₄×C₆₇	C₂²×C₄	1	16	yes	complemented
112	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
113	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
114	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
115	C₄⋊C₄×C₆₇	—	2	16	no	—
116	C₂²⋊C₄×C₆₇	—	2	8	no	—
117	C₂²⋊C₄×C₆₇	—	2	32	no	—
118	C₂²⋊C₄×C₆₇	C₂⁴	1	16	yes	—
119	C₄²×C₆₇	—	2	16	no	—
120	C₂³×C₆₇	—	2	4	no	—
121	C₂³×C₆₇	—	2	4	no	—
122	C₂³×C₆₇	—	2	32	no	—
123	C₂³×C₆₇	—	4	16	no	—
124	C₂³×C₆₇	—	4	8	no	—
125	C₂³×C₆₇	—	4	2	no	—
126	C₂³×C₆₇	—	4	8	no	—
127	C₂³×C₆₇	—	8	16	no	—
128	C₂³×C₆₇	—	8	4	no	—
129	C₂³×C₆₇	—	8	8	no	—
130	C₂³×C₆₇	—	8	4	no	—
131	C₂³×C₆₇	—	8	16	no	—
132	C₂³×C₆₇	—	8	16	no	—
133	C₂³×C₆₇	C₂×C₂²⋊C₄	1	8	yes	—
134	C₂³×C₆₇	C₂×C₂²⋊C₄	1	8	yes	—
135	C₂³×C₆₇	C₂×C₂²⋊C₄	1	16	yes	—
136	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
137	C₂³×C₆₇	C₂³×C₄	1	8	yes	—
138	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
139	C₂³×C₆₇	C₂³×C₄	1	8	yes	—
140	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
141	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
142	C₂³×C₆₇	C₂²×D₄	1	16	yes	—
143	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
144	Q₈×C₆₇	—	2	16	no	—
145	D₄×C₆₇	—	2	16	no	—
146	D₄×C₆₇	—	2	64	no	—
147	D₄×C₆₇	—	2	16	no	—
148	D₄×C₆₇	—	2	16	no	—
149	D₄×C₆₇	C₂×C₂²⋊C₄	1	64	yes	—
150	C₂×C₄×C₆₇	—	2	4	no	—
151	C₂×C₄×C₆₇	—	2	32	no	—
152	C₂×C₄×C₆₇	—	2	4	no	—
153	C₂×C₄×C₆₇	—	2	32	no	—
154	C₂×C₄×C₆₇	—	2	16	no	—
155	C₂×C₄×C₆₇	—	4	16	no	—
156	C₂×C₄×C₆₇	—	8	32	no	—
157	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
158	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	8	yes	—
159	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
160	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
161	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
162	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
163	C₂²×C₆₇	—	2	4	no	—
164	C₂²×C₆₇	—	2	2	no	—
165	C₂²×C₆₇	—	2	16	no	—
166	C₂²×C₆₇	—	2	8	no	—
167	C₂²×C₆₇	—	4	16	no	—
168	C₂²×C₆₇	—	4	8	no	—
169	C₂²×C₆₇	—	4	1	no	—
170	C₂²×C₆₇	—	4	8	no	—
171	C₂²×C₆₇	—	4	2	no	—
172	C₂²×C₆₇	—	4	1	no	—
173	C₂²×C₆₇	—	4	16	no	—
174	C₂²×C₆₇	—	4	8	no	—
175	C₂²×C₆₇	—	4	8	no	—
176	C₂²×C₆₇	—	4	8	no	—
177	C₂²×C₆₇	—	8	32	no	—
178	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
179	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	16	yes	—
180	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	2	yes	—
181	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
182	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
183	C₂²×C₆₇	C₂³×D₄	1	2	yes	—
184	C₄×C₆₇	—	2	8	no	—
185	C₄×C₆₇	—	4	16	no	—
186	C₄×C₆₇	C₂²×C₂²⋊C₄	1	8	yes	—
187	C₂×#2883	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
188	C₂×C₆₇	—	2	2	no	—
189	C₂×C₆₇	—	2	1	no	—
190	C₂×C₆₇	—	2	1	no	—
191	C₂×C₆₇	—	2	2	no	—
192	C₂×C₆₇	—	2	16	no	—
193	C₂×C₆₇	—	4	8	no	—
194	C₂×C₆₇	#2883	1	2	yes	direct product factor
195	C₂×C₆₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
196	C₂²×H(5,𝔽₂)	C₂×C₆₇	1	1	yes	characteristic
197	C₂×#518	C₂×C₆₇	1	2	yes	complemented
198	#2883	C₂×C₆₇	1	16	yes	direct product factor
199	C₂×#410	C₂×C₆₇	1	4	yes	complemented
200	C₂²×#99	C₂×C₆₇	1	8	yes	complemented
201	C₆₇	C₂×#2883	1	1	yes	characteristic, direct product factor, Sylow
202	C₂×H(5,𝔽₂)	—	2	8	no	—
203	C₂×H(5,𝔽₂)	—	2	8	no	—
204	C₂×H(5,𝔽₂)	C₄×C₆₇	1	8	yes	complemented
205	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	8	yes	—
206	C₂²×#41	C₄×C₆₇	1	4	yes	complemented
207	C₂²×#41	C₂²×C₆₇	1	2	yes	—
208	C₂³×D₄	C₄×C₆₇	1	4	yes	complemented
209	C₂³×D₄	C₂²×C₆₇	1	4	yes	—
210	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
211	#518	—	2	8	no	—
212	#518	C₂²×C₆₇	1	16	yes	complemented
213	C₂×C₄×D₄	—	2	8	no	—
214	C₂×#117	C₂²×C₆₇	1	4	yes	complemented
215	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	complemented
216	#410	C₂²×C₆₇	1	32	yes	complemented
217	C₂×#99	C₂²×C₆₇	1	16	yes	complemented
218	C₂×#99	C₂²×C₆₇	1	64	yes	complemented
219	C₂⁵	C₂×C₄×C₆₇	1	4	yes	complemented
220	C₂⁵	C₂³×C₆₇	1	2	yes	—
221	H(5,𝔽₂)	—	2	64	no	—
222	H(5,𝔽₂)	—	2	64	no	—
223	C₂×#41	—	2	16	no	—
224	C₂×#41	—	2	16	no	—
225	C₂×#41	—	2	8	no	—
226	C₂×#41	—	2	8	no	—
227	C₂×#41	C₂×C₄×C₆₇	1	8	yes	complemented
228	C₂×#41	C₂×C₄×C₆₇	1	32	yes	complemented
229	C₂×#41	C₂³×C₆₇	1	8	yes	—
230	C₂²×Q₈	C₂×C₄×C₆₇	1	2	yes	complemented
231	C₂²×D₄	—	2	16	no	—
232	C₂²×D₄	—	2	16	no	—
233	C₂²×D₄	—	2	16	no	—
234	C₂²×D₄	—	2	16	no	—
235	C₂²×D₄	C₂×C₄×C₆₇	1	4	yes	—
236	C₂²×D₄	C₂×C₄×C₆₇	1	4	yes	—
237	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	complemented
238	C₂²×D₄	C₂×C₄×C₆₇	1	8	yes	complemented
239	C₂²×D₄	C₂×C₄×C₆₇	1	16	yes	complemented
240	C₂²×D₄	C₂×C₄×C₆₇	1	32	yes	complemented
241	C₂²×D₄	D₄×C₆₇	1	8	yes	—
242	C₂²×D₄	D₄×C₆₇	1	8	yes	—
243	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
244	C₂²×D₄	C₂³×C₆₇	1	16	yes	—
245	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
246	C₂³×C₄	—	2	8	no	—
247	C₂³×C₄	C₂×C₄×C₆₇	1	4	yes	complemented
248	C₂³×C₄	C₂×C₄×C₆₇	1	4	yes	complemented
249	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
250	C₄×D₄	—	2	64	no	—
251	#117	—	2	8	no	—
252	#117	C₂³×C₆₇	1	16	yes	complemented
253	C₂×C₄⋊C₄	—	2	4	no	—
254	C₂×C₂²⋊C₄	—	2	8	no	—
255	C₂×C₂²⋊C₄	—	2	16	no	—
256	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
257	C₂×C₂²⋊C₄	C₂³×C₆₇	1	32	yes	complemented
258	C₂×C₄²	—	2	4	no	—
259	#99	C₂³×C₆₇	1	64	yes	complemented
260	C₂⁴	—	2	8	no	—
261	C₂⁴	—	2	8	no	—
262	C₂⁴	—	8	4	no	—
263	C₂⁴	—	8	8	no	—
264	C₂⁴	C₂²×C₄×C₆₇	1	4	yes	—
265	C₂⁴	C₂²×C₄×C₆₇	1	4	yes	—
266	C₂⁴	C₂²×C₄×C₆₇	1	4	yes	—
267	C₂⁴	C₂²×C₄×C₆₇	1	16	yes	complemented
268	C₂⁴	C₂×D₄×C₆₇	1	8	yes	—
269	C₂⁴	C₂×D₄×C₆₇	1	8	yes	—
270	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
271	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
272	#41	—	2	128	no	—
273	#41	—	2	32	no	—
274	#41	—	2	32	no	—
275	C₂×Q₈	—	2	4	no	—
276	C₂×Q₈	—	2	4	no	—
277	C₂×Q₈	C₂²×C₄×C₆₇	1	8	yes	complemented
278	C₂×D₄	—	2	16	no	—
279	C₂×D₄	—	2	16	no	—
280	C₂×D₄	—	2	64	no	—
281	C₂×D₄	—	2	8	no	—
282	C₂×D₄	—	2	8	no	—
283	C₂×D₄	—	2	64	no	—
284	C₂×D₄	—	2	128	no	—
285	C₂×D₄	—	2	4	no	—
286	C₂×D₄	—	2	4	no	—
287	C₂×D₄	C₂²⋊C₄×C₆₇	1	16	yes	—
288	C₂×D₄	C₂²⋊C₄×C₆₇	1	16	yes	—
289	C₂×D₄	C₂²⋊C₄×C₆₇	1	32	yes	—
290	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
291	C₂×D₄	C₂²×C₄×C₆₇	1	16	yes	—
292	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	complemented
293	C₂×D₄	C₂²×C₄×C₆₇	1	32	yes	complemented
294	C₂×D₄	C₂×D₄×C₆₇	1	32	yes	—
295	C₂×D₄	C₂⁴×C₆₇	1	8	yes	—
296	C₂²×C₄	—	2	8	no	—
297	C₂²×C₄	—	2	8	no	—
298	C₂²×C₄	—	2	8	no	—
299	C₂²×C₄	—	2	8	no	—
300	C₂²×C₄	—	2	8	no	—
301	C₂²×C₄	—	2	32	no	—
302	C₂²×C₄	—	8	16	no	—
303	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
304	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
305	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
306	C₂²×C₄	C₂²×C₄×C₆₇	1	16	yes	complemented
307	C₂²×C₄	C₂²×C₄×C₆₇	1	16	yes	complemented
308	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
309	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
310	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
311	C₄⋊C₄	—	2	16	no	—
312	C₂²⋊C₄	—	2	8	no	—
313	C₂²⋊C₄	—	2	32	no	—
314	C₂²⋊C₄	C₂⁴×C₆₇	1	16	yes	—
315	C₄²	—	2	16	no	—
316	C₂³	—	2	4	no	—
317	C₂³	—	2	4	no	—
318	C₂³	—	2	32	no	—
319	C₂³	—	4	16	no	—
320	C₂³	—	4	8	no	—
321	C₂³	—	4	2	no	—
322	C₂³	—	4	8	no	—
323	C₂³	—	8	16	no	—
324	C₂³	—	8	4	no	—
325	C₂³	—	8	8	no	—
326	C₂³	—	8	4	no	—
327	C₂³	—	8	16	no	—
328	C₂³	—	8	16	no	—
329	C₂³	C₂×C₂²⋊C₄×C₆₇	1	8	yes	—
330	C₂³	C₂×C₂²⋊C₄×C₆₇	1	8	yes	—
331	C₂³	C₂×C₂²⋊C₄×C₆₇	1	16	yes	—
332	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
333	C₂³	C₂³×C₄×C₆₇	1	8	yes	—
334	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
335	C₂³	C₂³×C₄×C₆₇	1	8	yes	—
336	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
337	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
338	C₂³	C₂²×D₄×C₆₇	1	16	yes	—
339	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
340	Q₈	—	2	16	no	—
341	D₄	—	2	16	no	—
342	D₄	—	2	64	no	—
343	D₄	—	2	16	no	—
344	D₄	—	2	16	no	—
345	D₄	C₂×C₂²⋊C₄×C₆₇	1	64	yes	—
346	C₂×C₄	—	2	4	no	—
347	C₂×C₄	—	2	4	no	—
348	C₂×C₄	—	2	32	no	—
349	C₂×C₄	—	2	32	no	—
350	C₂×C₄	—	2	16	no	—
351	C₂×C₄	—	4	16	no	—
352	C₂×C₄	—	8	32	no	—
353	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
354	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
355	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	8	yes	—
356	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
357	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
358	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
359	C₂²	—	2	4	no	—
360	C₂²	—	2	2	no	—
361	C₂²	—	2	16	no	—
362	C₂²	—	2	8	no	—
363	C₂²	—	4	16	no	—
364	C₂²	—	4	8	no	—
365	C₂²	—	4	1	no	—
366	C₂²	—	4	8	no	—
367	C₂²	—	4	2	no	—
368	C₂²	—	4	1	no	—
369	C₂²	—	4	16	no	—
370	C₂²	—	4	8	no	—
371	C₂²	—	4	8	no	—
372	C₂²	—	4	8	no	—
373	C₂²	—	8	32	no	—
374	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
375	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	16	yes	—
376	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	2	yes	—
377	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
378	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
379	C₂²	C₂³×D₄×C₆₇	1	2	yes	—
380	C₄	—	2	8	no	—
381	C₄	—	4	16	no	—
382	C₄	C₂²×C₂²⋊C₄×C₆₇	1	8	yes	—
383	C₂	—	2	2	no	—
384	C₂	—	2	1	no	—
385	C₂	—	2	1	no	—
386	C₂	—	2	2	no	—
387	C₂	—	2	16	no	—
388	C₂	—	4	8	no	—
389	C₂	C₆₇×#2883	1	2	yes	direct product factor
390	C₂	C₂³×C₂²⋊C₄×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×C₂²⋊C₄
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂²×H(5,𝔽₂)×C₆₇	C₂
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂³×D₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#518×C₆₇	C₂	semidirect
3. C₆₇×#2883	C₂	direct (N × H)
4. C₂×#410×C₆₇	C₂	semidirect
5. C₂²×#99×C₆₇	C₂	semidirect
8. C₂×H(5,𝔽₂)×C₆₇	C₄	semidirect
10. C₂²×#41×C₆₇	C₄	semidirect
12. C₂³×D₄×C₆₇	C₄	semidirect
16. #518×C₆₇	C₂²	semidirect
18. C₂×#117×C₆₇	C₂²	semidirect
19. C₂²×C₂²⋊C₄×C₆₇	C₂²	semidirect
20. #410×C₆₇	C₂²	semidirect
21. C₂×#99×C₆₇	C₂²	semidirect
22. C₂×#99×C₆₇	C₂²	semidirect
23. C₂⁵×C₆₇	C₂×C₄	semidirect
31. C₂×#41×C₆₇	C₂×C₄	semidirect
32. C₂×#41×C₆₇	C₂×C₄	semidirect
34. C₂²×Q₈×C₆₇	C₂×C₄	semidirect
41. C₂²×D₄×C₆₇	C₂×C₄	semidirect
42. C₂²×D₄×C₆₇	C₂×C₄	semidirect
43. C₂²×D₄×C₆₇	C₂×C₄	semidirect
44. C₂²×D₄×C₆₇	C₂×C₄	semidirect
51. C₂³×C₄×C₆₇	C₂×C₄	semidirect
52. C₂³×C₄×C₆₇	C₂×C₄	semidirect
56. #117×C₆₇	C₂³	semidirect
61. C₂×C₂²⋊C₄×C₆₇	C₂³	semidirect
63. #99×C₆₇	C₂³	semidirect
71. C₂⁴×C₆₇	C₂²×C₄	semidirect
81. C₂×Q₈×C₆₇	C₂²×C₄	semidirect
96. C₂×D₄×C₆₇	C₂²×C₄	semidirect
97. C₂×D₄×C₆₇	C₂²×C₄	semidirect
110. C₂²×C₄×C₆₇	C₂²×C₄	semidirect
111. C₂²×C₄×C₆₇	C₂²×C₄	semidirect
187. C₂×#2883	C₆₇	direct (N × H)
194. C₂×C₆₇	#2883	direct (N × H)
197. C₂×#518	C₂×C₆₇	semidirect
198. #2883	C₂×C₆₇	direct (N × H)
199. C₂×#410	C₂×C₆₇	semidirect
200. C₂²×#99	C₂×C₆₇	semidirect
201. C₆₇	C₂×#2883	direct (N × H)
204. C₂×H(5,𝔽₂)	C₄×C₆₇	semidirect
206. C₂²×#41	C₄×C₆₇	semidirect
208. C₂³×D₄	C₄×C₆₇	semidirect
212. #518	C₂²×C₆₇	semidirect
214. C₂×#117	C₂²×C₆₇	semidirect
215. C₂²×C₂²⋊C₄	C₂²×C₆₇	semidirect
216. #410	C₂²×C₆₇	semidirect
217. C₂×#99	C₂²×C₆₇	semidirect
218. C₂×#99	C₂²×C₆₇	semidirect
219. C₂⁵	C₂×C₄×C₆₇	semidirect
227. C₂×#41	C₂×C₄×C₆₇	semidirect
228. C₂×#41	C₂×C₄×C₆₇	semidirect
230. C₂²×Q₈	C₂×C₄×C₆₇	semidirect
237. C₂²×D₄	C₂×C₄×C₆₇	semidirect
238. C₂²×D₄	C₂×C₄×C₆₇	semidirect
239. C₂²×D₄	C₂×C₄×C₆₇	semidirect
240. C₂²×D₄	C₂×C₄×C₆₇	semidirect
247. C₂³×C₄	C₂×C₄×C₆₇	semidirect
248. C₂³×C₄	C₂×C₄×C₆₇	semidirect
252. #117	C₂³×C₆₇	semidirect
257. C₂×C₂²⋊C₄	C₂³×C₆₇	semidirect
259. #99	C₂³×C₆₇	semidirect
267. C₂⁴	C₂²×C₄×C₆₇	semidirect
277. C₂×Q₈	C₂²×C₄×C₆₇	semidirect
292. C₂×D₄	C₂²×C₄×C₆₇	semidirect
293. C₂×D₄	C₂²×C₄×C₆₇	semidirect
306. C₂²×C₄	C₂²×C₄×C₆₇	semidirect
307. C₂²×C₄	C₂²×C₄×C₆₇	semidirect
389. C₂	C₆₇×#2883	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×H(5,𝔽₂)×C₆₇	2
C₂×#518×C₆₇	2
C₆₇×#2883	2
C₂×#410×C₆₇	2
C₂²×#99×C₆₇	2
C₂×#2883	67

Subgroups — C2×C67×#2883 #30739319

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2883 #30739319