Subgroups — C₂×C₆₇×#3380 #30740604

Order 17152.

1918 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#554×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3380	C₂	1	16	yes	maximal, direct product factor
3	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#499×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#478×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₂×#476×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×D₈×C₆₇	C₂	1	2	yes	maximal
10	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
11	#554×C₆₇	—	2	8	no	—
12	#554×C₆₇	C₂²	1	16	yes	complemented
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
14	C₂×#125×C₆₇	C₂²	1	4	yes	—
15	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
16	C₂×C₂²⋊Q₈×C₆₇	—	2	4	no	—
17	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
20	C₂²×C₄⋊C₄×C₆₇	—	2	2	no	—
21	#500×C₆₇	C₂²	1	8	yes	complemented
22	#499×C₆₇	C₂²	1	16	yes	complemented
23	#478×C₆₇	C₂²	1	32	yes	complemented
24	#476×C₆₇	C₂²	1	16	yes	complemented
25	C₈×Q₈×C₆₇	C₂²	1	8	yes	complemented
26	C₄×D₈×C₆₇	C₂²	1	16	yes	—
27	C₄×D₈×C₆₇	C₂²	1	8	yes	—
28	C₂×#107×C₆₇	C₂²	1	2	yes	—
29	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×#107×C₆₇	C₂²	1	2	yes	—
31	C₂×#106×C₆₇	C₂²	1	4	yes	—
32	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
34	C₂×#102×C₆₇	C₂²	1	4	yes	—
35	C₂×#102×C₆₇	C₂²	1	2	yes	—
36	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
37	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
38	C₂²×Q₈×C₆₇	—	2	2	no	—
39	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
40	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
41	C₂³×C₄×C₆₇	—	2	2	no	—
42	C₂³×C₄×C₆₇	—	2	4	no	—
43	C₂×D₈×C₆₇	Q₈	1	8	yes	complemented
44	C₂×D₈×C₆₇	C₂³	1	4	yes	—
45	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
46	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
48	C₄⋊Q₈×C₆₇	—	2	4	no	—
49	C₄⋊Q₈×C₆₇	C₂³	1	8	yes	—
50	#125×C₆₇	—	2	8	no	—
51	#125×C₆₇	C₂³	1	16	yes	—
52	C₂²⋊Q₈×C₆₇	—	2	16	no	—
53	C₂²⋊Q₈×C₆₇	—	2	32	no	—
54	C₄×Q₈×C₆₇	D₄	1	4	yes	—
55	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
56	C₄×D₄×C₆₇	—	2	8	no	—
57	C₄×D₄×C₆₇	—	2	4	no	—
58	C₄×D₄×C₆₇	C₂³	1	8	yes	—
59	C₄×D₄×C₆₇	C₂³	1	16	yes	—
60	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
61	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
63	C₂×C₄⋊C₄×C₆₇	—	2	16	no	—
64	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
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	4	yes	—
68	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
70	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
71	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
72	#107×C₆₇	C₂³	1	8	yes	—
73	#107×C₆₇	C₂³	1	4	yes	—
74	#107×C₆₇	C₂³	1	8	yes	—
75	#106×C₆₇	C₂³	1	16	yes	—
76	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
77	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
78	#102×C₆₇	C₂³	1	16	yes	—
79	#102×C₆₇	C₂³	1	8	yes	—
80	C₄×C₈×C₆₇	C₂³	1	8	yes	—
81	C₄×C₈×C₆₇	C₂³	1	4	yes	—
82	C₂⁴×C₆₇	—	2	2	no	—
83	C₂×Q₈×C₆₇	—	2	4	no	—
84	C₂×Q₈×C₆₇	—	2	8	no	—
85	C₂×Q₈×C₆₇	—	8	4	no	—
86	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
87	C₂×D₄×C₆₇	—	2	2	no	—
88	C₂×D₄×C₆₇	—	2	4	no	—
89	C₂×D₄×C₆₇	C₂×Q₈	1	8	yes	—
90	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
91	C₂²×C₄×C₆₇	—	2	2	no	—
92	C₂²×C₄×C₆₇	—	2	4	no	—
93	C₂²×C₄×C₆₇	—	2	8	no	—
94	C₂²×C₄×C₆₇	—	2	16	no	—
95	C₂²×C₄×C₆₇	—	4	8	no	—
96	C₂²×C₄×C₆₇	—	8	8	no	—
97	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
98	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
100	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
102	D₈×C₆₇	C₂×Q₈	1	16	yes	complemented
103	C₂×C₈×C₆₇	—	2	4	no	—
104	C₂×C₈×C₆₇	—	2	2	no	—
105	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
106	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
107	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
108	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
109	C₄⋊C₄×C₆₇	—	2	16	no	—
110	C₄⋊C₄×C₆₇	—	2	8	no	—
111	C₄⋊C₄×C₆₇	—	2	16	no	—
112	C₄⋊C₄×C₆₇	—	2	2	no	—
113	C₄⋊C₄×C₆₇	—	2	4	no	—
114	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
115	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
116	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
117	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
118	C₂²⋊C₄×C₆₇	—	2	8	no	—
119	C₂²⋊C₄×C₆₇	—	2	16	no	—
120	C₄²×C₆₇	C₂×D₄	1	4	yes	—
121	C₄²×C₆₇	C₂×D₄	1	2	yes	—
122	C₄²×C₆₇	C₂⁴	1	4	yes	—
123	C₄²×C₆₇	C₂⁴	1	2	yes	—
124	C₂³×C₆₇	—	2	8	no	—
125	C₂³×C₆₇	—	2	4	no	—
126	C₂³×C₆₇	—	4	4	no	—
127	C₂³×C₆₇	—	4	4	no	—
128	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
129	Q₈×C₆₇	—	2	8	no	—
130	Q₈×C₆₇	—	8	8	no	—
131	D₄×C₆₇	—	2	8	no	—
132	D₄×C₆₇	—	2	4	no	—
133	D₄×C₆₇	C₂²×Q₈	1	8	yes	—
134	C₂×C₄×C₆₇	—	2	4	no	—
135	C₂×C₄×C₆₇	—	2	4	no	—
136	C₂×C₄×C₆₇	—	2	2	no	—
137	C₂×C₄×C₆₇	—	2	4	no	—
138	C₂×C₄×C₆₇	—	2	8	no	—
139	C₂×C₄×C₆₇	—	4	4	no	—
140	C₂×C₄×C₆₇	—	4	8	no	—
141	C₂×C₄×C₆₇	—	4	16	no	—
142	C₂×C₄×C₆₇	—	8	16	no	—
143	C₂×C₄×C₆₇	#135	1	4	yes	—
144	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
145	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
146	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
147	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
149	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
150	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
151	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
152	C₈×C₆₇	—	2	4	no	—
153	C₈×C₆₇	—	2	2	no	—
154	C₈×C₆₇	C₂²×Q₈	1	4	yes	—
155	C₂²×C₆₇	—	2	8	no	—
156	C₂²×C₆₇	—	4	16	no	—
157	C₂²×C₆₇	—	4	4	no	—
158	C₂²×C₆₇	—	4	4	no	—
159	C₂²×C₆₇	D₄×Q₈	1	2	yes	—
160	C₂²×C₆₇	C₂×#135	1	2	yes	—
161	C₂²×C₆₇	#576	1	2	yes	—
162	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
163	C₄×C₆₇	—	2	4	no	—
164	C₄×C₆₇	—	2	4	no	—
165	C₄×C₆₇	—	2	2	no	—
166	C₄×C₆₇	—	4	4	no	—
167	C₄×C₆₇	—	4	8	no	—
168	C₄×C₆₇	D₄×Q₈	1	2	yes	—
169	C₄×C₆₇	C₂×#135	1	4	yes	—
170	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
171	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
172	C₂×#3380	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
173	C₂×C₆₇	—	4	8	no	—
174	C₂×C₆₇	#3380	1	4	yes	direct product factor
175	C₂×C₆₇	C₂×D₄×Q₈	1	1	yes	characteristic
176	C₂×C₆₇	C₂²×#135	1	1	yes	characteristic
177	C₂×C₆₇	C₂×#576	1	1	yes	characteristic
178	C₂×#554	C₂×C₆₇	1	2	yes	complemented
179	#3380	C₂×C₆₇	1	16	yes	direct product factor
180	C₂×#500	C₂×C₆₇	1	1	yes	characteristic, complemented
181	C₂×#499	C₂×C₆₇	1	2	yes	complemented
182	C₂×#478	C₂×C₆₇	1	4	yes	complemented
183	C₂×#476	C₂×C₆₇	1	2	yes	complemented
184	C₂×C₈×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
185	C₂×C₄×D₈	C₂×C₆₇	1	2	yes	—
186	C₂×C₄×D₈	C₂×C₆₇	1	1	yes	characteristic
187	C₆₇	C₂×#3380	1	1	yes	characteristic, direct product factor, Sylow
188	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
189	#554	—	2	8	no	—
190	#554	C₂²×C₆₇	1	16	yes	complemented
191	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
192	C₂×#125	C₂²×C₆₇	1	4	yes	—
193	C₂×C₂²⋊Q₈	—	2	2	no	—
194	C₂×C₂²⋊Q₈	—	2	4	no	—
195	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
196	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
197	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
198	C₂²×C₄⋊C₄	—	2	2	no	—
199	#500	C₂²×C₆₇	1	8	yes	complemented
200	#499	C₂²×C₆₇	1	16	yes	complemented
201	#478	C₂²×C₆₇	1	32	yes	complemented
202	#476	C₂²×C₆₇	1	16	yes	complemented
203	C₈×Q₈	C₂²×C₆₇	1	8	yes	complemented
204	C₄×D₈	C₂²×C₆₇	1	16	yes	—
205	C₄×D₈	C₂²×C₆₇	1	8	yes	—
206	C₂×#107	C₂²×C₆₇	1	2	yes	—
207	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
208	C₂×#107	C₂²×C₆₇	1	2	yes	—
209	C₂×#106	C₂²×C₆₇	1	4	yes	—
210	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
211	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
212	C₂×#102	C₂²×C₆₇	1	2	yes	—
213	C₂×#102	C₂²×C₆₇	1	4	yes	—
214	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
215	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
216	C₂²×Q₈	—	2	2	no	—
217	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
218	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
219	C₂³×C₄	—	2	2	no	—
220	C₂³×C₄	—	2	4	no	—
221	C₂×D₈	Q₈×C₆₇	1	8	yes	complemented
222	C₂×D₈	C₂³×C₆₇	1	4	yes	—
223	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
224	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
225	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
226	C₄⋊Q₈	—	2	4	no	—
227	C₄⋊Q₈	C₂³×C₆₇	1	8	yes	—
228	#125	—	2	8	no	—
229	#125	C₂³×C₆₇	1	16	yes	—
230	C₂²⋊Q₈	—	2	16	no	—
231	C₂²⋊Q₈	—	2	32	no	—
232	C₄×Q₈	D₄×C₆₇	1	4	yes	—
233	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
234	C₄×D₄	—	2	8	no	—
235	C₄×D₄	—	2	4	no	—
236	C₄×D₄	C₂³×C₆₇	1	8	yes	—
237	C₄×D₄	C₂³×C₆₇	1	16	yes	—
238	C₂×C₄⋊C₄	—	2	4	no	—
239	C₂×C₄⋊C₄	—	2	2	no	—
240	C₂×C₄⋊C₄	—	2	4	no	—
241	C₂×C₄⋊C₄	—	2	16	no	—
242	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
243	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
244	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
245	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
246	C₂×C₂²⋊C₄	—	2	2	no	—
247	C₂×C₂²⋊C₄	—	2	4	no	—
248	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
249	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
250	#107	C₂³×C₆₇	1	4	yes	—
251	#107	C₂³×C₆₇	1	8	yes	—
252	#107	C₂³×C₆₇	1	8	yes	—
253	#106	C₂³×C₆₇	1	16	yes	—
254	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
255	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
256	#102	C₂³×C₆₇	1	16	yes	—
257	#102	C₂³×C₆₇	1	8	yes	—
258	C₄×C₈	C₂³×C₆₇	1	8	yes	—
259	C₄×C₈	C₂³×C₆₇	1	4	yes	—
260	C₂⁴	—	2	2	no	—
261	C₂×Q₈	—	2	4	no	—
262	C₂×Q₈	—	2	8	no	—
263	C₂×Q₈	—	8	4	no	—
264	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
265	C₂×D₄	—	2	2	no	—
266	C₂×D₄	—	2	4	no	—
267	C₂×D₄	C₂×Q₈×C₆₇	1	8	yes	—
268	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	2	4	no	—
271	C₂²×C₄	—	2	8	no	—
272	C₂²×C₄	—	2	16	no	—
273	C₂²×C₄	—	4	8	no	—
274	C₂²×C₄	—	8	8	no	—
275	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
276	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
277	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
278	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
279	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
280	D₈	C₂×Q₈×C₆₇	1	16	yes	complemented
281	C₂×C₈	—	2	4	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
284	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
285	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
286	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
287	C₄⋊C₄	—	2	16	no	—
288	C₄⋊C₄	—	2	8	no	—
289	C₄⋊C₄	—	2	16	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	4	no	—
292	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
293	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
294	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
295	C₄⋊C₄	C₂⁴×C₆₇	1	8	yes	—
296	C₂²⋊C₄	—	2	8	no	—
297	C₂²⋊C₄	—	2	16	no	—
298	C₄²	C₂×D₄×C₆₇	1	4	yes	—
299	C₄²	C₂×D₄×C₆₇	1	2	yes	—
300	C₄²	C₂⁴×C₆₇	1	4	yes	—
301	C₄²	C₂⁴×C₆₇	1	2	yes	—
302	C₂³	—	2	8	no	—
303	C₂³	—	2	4	no	—
304	C₂³	—	4	4	no	—
305	C₂³	—	4	4	no	—
306	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
307	Q₈	—	2	8	no	—
308	Q₈	—	8	8	no	—
309	D₄	—	2	8	no	—
310	D₄	—	2	4	no	—
311	D₄	C₂²×Q₈×C₆₇	1	8	yes	—
312	C₂×C₄	—	2	4	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	2	8	no	—
317	C₂×C₄	—	4	4	no	—
318	C₂×C₄	—	4	8	no	—
319	C₂×C₄	—	4	16	no	—
320	C₂×C₄	—	8	16	no	—
321	C₂×C₄	#135×C₆₇	1	4	yes	—
322	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
323	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
324	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
325	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
326	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
327	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
328	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
329	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
330	C₈	—	2	4	no	—
331	C₈	—	2	2	no	—
332	C₈	C₂²×Q₈×C₆₇	1	4	yes	—
333	C₂²	—	2	8	no	—
334	C₂²	—	4	4	no	—
335	C₂²	—	4	4	no	—
336	C₂²	—	4	16	no	—
337	C₂²	D₄×Q₈×C₆₇	1	2	yes	—
338	C₂²	C₂×#135×C₆₇	1	2	yes	—
339	C₂²	#576×C₆₇	1	2	yes	—
340	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
341	C₄	—	2	4	no	—
342	C₄	—	2	4	no	—
343	C₄	—	2	2	no	—
344	C₄	—	4	4	no	—
345	C₄	—	4	8	no	—
346	C₄	D₄×Q₈×C₆₇	1	2	yes	—
347	C₄	C₂×#135×C₆₇	1	4	yes	—
348	C₄	C₂³×Q₈×C₆₇	1	1	yes	characteristic
349	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
350	C₂	—	4	8	no	—
351	C₂	C₆₇×#3380	1	4	yes	direct product factor
352	C₂	C₂×D₄×Q₈×C₆₇	1	1	yes	characteristic
353	C₂	C₂²×#135×C₆₇	1	1	yes	characteristic
354	C₂	C₂×#576×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂²×D₈×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#554×C₆₇	C₂	semidirect
2. C₆₇×#3380	C₂	direct (N × H)
3. C₂×#500×C₆₇	C₂	semidirect
4. C₂×#499×C₆₇	C₂	semidirect
5. C₂×#478×C₆₇	C₂	semidirect
6. C₂×#476×C₆₇	C₂	semidirect
7. C₂×C₈×Q₈×C₆₇	C₂	semidirect
12. #554×C₆₇	C₂²	semidirect
21. #500×C₆₇	C₂²	semidirect
22. #499×C₆₇	C₂²	semidirect
23. #478×C₆₇	C₂²	semidirect
24. #476×C₆₇	C₂²	semidirect
25. C₈×Q₈×C₆₇	C₂²	semidirect
43. C₂×D₈×C₆₇	Q₈	semidirect
102. D₈×C₆₇	C₂×Q₈	semidirect
172. C₂×#3380	C₆₇	direct (N × H)
174. C₂×C₆₇	#3380	direct (N × H)
178. C₂×#554	C₂×C₆₇	semidirect
179. #3380	C₂×C₆₇	direct (N × H)
180. C₂×#500	C₂×C₆₇	semidirect
181. C₂×#499	C₂×C₆₇	semidirect
182. C₂×#478	C₂×C₆₇	semidirect
183. C₂×#476	C₂×C₆₇	semidirect
184. C₂×C₈×Q₈	C₂×C₆₇	semidirect
187. C₆₇	C₂×#3380	direct (N × H)
190. #554	C₂²×C₆₇	semidirect
199. #500	C₂²×C₆₇	semidirect
200. #499	C₂²×C₆₇	semidirect
201. #478	C₂²×C₆₇	semidirect
202. #476	C₂²×C₆₇	semidirect
203. C₈×Q₈	C₂²×C₆₇	semidirect
221. C₂×D₈	Q₈×C₆₇	semidirect
280. D₈	C₂×Q₈×C₆₇	semidirect
351. C₂	C₆₇×#3380	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#554×C₆₇	2
C₆₇×#3380	2
C₂×#500×C₆₇	2
C₂×#499×C₆₇	2
C₂×#478×C₆₇	2
C₂×#476×C₆₇	2
C₂×C₈×Q₈×C₆₇	2
C₂×C₄×D₈×C₆₇	2
C₂×#3380	67

Subgroups — C2×C67×#3380 #30740604

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3380 #30740604