Subgroups — C₂²×#380×C₆₇ #30713255

Order 17152.

8670 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊C₄×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂³×C₂²⋊C₄×C₆₇	C₂	1	2	yes	maximal
4	C₂×#380×C₆₇	C₂	1	24	yes	maximal, direct product factor
5	C₂²×#95×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂⁶×C₆₇	C₂²	1	1	yes	characteristic
7	C₂⁴×C₄×C₆₇	—	2	16	no	—
8	C₂⁴×C₄×C₆₇	C₄	1	16	yes	complemented
9	C₂⁴×C₄×C₆₇	C₂²	1	4	yes	—
10	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₆₇	C₂²	1	12	yes	—
12	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄⋊C₄×C₆₇	—	2	16	no	—
14	C₂²×C₄⋊C₄×C₆₇	C₂²	1	24	yes	complemented
15	C₂²×C₂²⋊C₄×C₆₇	—	2	16	no	—
16	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	24	yes	—
17	#380×C₆₇	C₂²	1	64	yes	direct product factor
18	C₂×#95×C₆₇	C₂²	1	24	yes	complemented
19	C₂⁵×C₆₇	—	2	8	no	—
20	C₂⁵×C₆₇	—	2	8	no	—
21	C₂⁵×C₆₇	C₂×C₄	1	8	yes	—
22	C₂⁵×C₆₇	D₄	1	8	yes	—
23	C₂⁵×C₆₇	Q₈	1	8	yes	—
24	C₂⁵×C₆₇	C₂³	1	6	yes	—
25	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
26	C₂³×C₄×C₆₇	—	2	8	no	—
27	C₂³×C₄×C₆₇	—	2	128	no	—
28	C₂³×C₄×C₆₇	—	2	96	no	—
29	C₂³×C₄×C₆₇	—	2	16	no	—
30	C₂³×C₄×C₆₇	—	2	16	no	—
31	C₂³×C₄×C₆₇	—	2	16	no	—
32	C₂³×C₄×C₆₇	C₂×C₄	1	8	yes	—
33	C₂³×C₄×C₆₇	C₂×C₄	1	8	yes	—
34	C₂³×C₄×C₆₇	C₂×C₄	1	96	yes	complemented
35	C₂³×C₄×C₆₇	D₄	1	16	yes	—
36	C₂³×C₄×C₆₇	C₂³	1	24	yes	—
37	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
38	C₂³×C₄×C₆₇	C₂³	1	16	yes	—
39	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
40	C₂×C₄⋊C₄×C₆₇	—	2	96	no	—
41	C₂×C₄⋊C₄×C₆₇	C₂³	1	32	yes	complemented
42	C₂×C₂²⋊C₄×C₆₇	—	2	96	no	—
43	C₂×C₂²⋊C₄×C₆₇	C₂³	1	32	yes	—
44	#95×C₆₇	C₂³	1	32	yes	complemented
45	C₂⁴×C₆₇	—	2	32	no	—
46	C₂⁴×C₆₇	—	2	32	no	—
47	C₂⁴×C₆₇	—	2	64	no	—
48	C₂⁴×C₆₇	—	2	64	no	—
49	C₂⁴×C₆₇	—	2	24	no	—
50	C₂⁴×C₆₇	—	2	24	no	—
51	C₂⁴×C₆₇	C₄⋊C₄	1	64	yes	complemented
52	C₂⁴×C₆₇	C₂²×C₄	1	24	yes	—
53	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
54	C₂⁴×C₆₇	C₂×D₄	1	24	yes	—
55	C₂⁴×C₆₇	C₂×D₄	1	8	yes	—
56	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
57	C₂⁴×C₆₇	C₂×Q₈	1	24	yes	—
58	C₂⁴×C₆₇	C₂×Q₈	1	4	yes	—
59	C₂⁴×C₆₇	#41	1	8	yes	—
60	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
61	C₂⁴×C₆₇	C₂⁴	1	3	yes	—
62	C₂²×C₄×C₆₇	—	2	32	no	—
63	C₂²×C₄×C₆₇	—	2	24	no	—
64	C₂²×C₄×C₆₇	—	2	384	no	—
65	C₂²×C₄×C₆₇	—	2	64	no	—
66	C₂²×C₄×C₆₇	—	2	48	no	—
67	C₂²×C₄×C₆₇	—	2	48	no	—
68	C₂²×C₄×C₆₇	—	2	48	no	—
69	C₂²×C₄×C₆₇	—	4	64	no	—
70	C₂²×C₄×C₆₇	C₂²⋊C₄	1	64	yes	complemented
71	C₂²×C₄×C₆₇	C₂²×C₄	1	24	yes	—
72	C₂²×C₄×C₆₇	C₂²×C₄	1	24	yes	—
73	C₂²×C₄×C₆₇	C₂²×C₄	1	64	yes	complemented
74	C₂²×C₄×C₆₇	C₂×D₄	1	48	yes	—
75	C₂²×C₄×C₆₇	C₂⁴	1	16	yes	—
76	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
77	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
78	C₄⋊C₄×C₆₇	—	2	64	no	—
79	C₂²⋊C₄×C₆₇	—	2	64	no	—
80	C₂³×C₆₇	—	2	48	no	—
81	C₂³×C₆₇	—	2	48	no	—
82	C₂³×C₆₇	—	2	96	no	—
83	C₂³×C₆₇	—	2	96	no	—
84	C₂³×C₆₇	—	2	8	no	—
85	C₂³×C₆₇	—	2	8	no	—
86	C₂³×C₆₇	—	2	256	no	—
87	C₂³×C₆₇	C₂×C₂²⋊C₄	1	16	yes	—
88	C₂³×C₆₇	C₂×C₄⋊C₄	1	16	yes	—
89	C₂³×C₆₇	C₂×C₄⋊C₄	1	96	yes	complemented
90	C₂³×C₆₇	#117	1	16	yes	—
91	C₂³×C₆₇	#121	1	32	yes	—
92	C₂³×C₆₇	C₂²⋊Q₈	1	32	yes	—
93	C₂³×C₆₇	C₂³×C₄	1	8	yes	—
94	C₂³×C₆₇	C₂³×C₄	1	6	yes	—
95	C₂³×C₆₇	C₂²×D₄	1	8	yes	—
96	C₂³×C₆₇	C₂²×D₄	1	12	yes	—
97	C₂³×C₆₇	C₂²×D₄	1	6	yes	—
98	C₂³×C₆₇	C₂²×Q₈	1	8	yes	—
99	C₂³×C₆₇	C₂²×Q₈	1	6	yes	—
100	C₂³×C₆₇	C₂×#41	1	12	yes	—
101	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
102	C₂×C₄×C₆₇	—	2	48	no	—
103	C₂×C₄×C₆₇	—	2	8	no	—
104	C₂×C₄×C₆₇	—	2	128	no	—
105	C₂×C₄×C₆₇	—	2	16	no	—
106	C₂×C₄×C₆₇	—	2	16	no	—
107	C₂×C₄×C₆₇	—	2	16	no	—
108	C₂×C₄×C₆₇	—	4	96	no	—
109	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	96	yes	complemented
110	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
111	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
112	C₂×C₄×C₆₇	C₂²×D₄	1	16	yes	—
113	C₂²×C₆₇	—	2	8	no	—
114	C₂²×C₆₇	—	2	8	no	—
115	C₂²×C₆₇	—	2	16	no	—
116	C₂²×C₆₇	—	2	16	no	—
117	C₂²×C₆₇	—	2	192	no	—
118	C₂²×C₆₇	#380	1	64	yes	direct product factor
119	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	12	yes	—
120	C₂²×C₆₇	C₂²×C₄⋊C₄	1	12	yes	—
121	C₂²×C₆₇	C₂²×C₄⋊C₄	1	16	yes	complemented
122	C₂²×C₆₇	C₂×#117	1	12	yes	—
123	C₂²×C₆₇	C₂×#121	1	24	yes	—
124	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	24	yes	—
125	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
126	C₂²×C₆₇	C₂³×D₄	1	2	yes	—
127	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
128	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
129	C₂²×C₆₇	C₂²×#41	1	2	yes	—
130	C₄×C₆₇	—	2	8	no	—
131	C₄×C₆₇	—	4	16	no	—
132	C₄×C₆₇	C₂²×C₂²⋊C₄	1	16	yes	complemented
133	C₂²×#380	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂×C₆₇	—	2	16	no	—
135	C₂×C₆₇	C₂×#380	1	24	yes	direct product factor
136	C₂×C₆₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₆₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₆₇	C₂²×#117	1	1	yes	characteristic
139	C₂×C₆₇	C₂²×#121	1	2	yes	—
140	C₂×C₆₇	C₂²×C₂²⋊Q₈	1	2	yes	—
141	C₂⁵×C₄	C₂×C₆₇	1	1	yes	characteristic
142	C₂³×C₄⋊C₄	C₂×C₆₇	1	2	yes	complemented
143	C₂³×C₂²⋊C₄	C₂×C₆₇	1	2	yes	—
144	C₂×#380	C₂×C₆₇	1	24	yes	direct product factor
145	C₂²×#95	C₂×C₆₇	1	2	yes	complemented
146	C₆₇	C₂²×#380	1	1	yes	characteristic, direct product factor, Sylow
147	C₂⁶	C₂²×C₆₇	1	1	yes	characteristic
148	C₂⁴×C₄	—	2	16	no	—
149	C₂⁴×C₄	C₄×C₆₇	1	16	yes	complemented
150	C₂⁴×C₄	C₂²×C₆₇	1	4	yes	—
151	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
152	C₂⁴×C₄	C₂²×C₆₇	1	12	yes	—
153	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
154	C₂²×C₄⋊C₄	—	2	16	no	—
155	C₂²×C₄⋊C₄	C₂²×C₆₇	1	24	yes	complemented
156	C₂²×C₂²⋊C₄	—	2	16	no	—
157	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	24	yes	—
158	#380	C₂²×C₆₇	1	64	yes	direct product factor
159	C₂×#95	C₂²×C₆₇	1	24	yes	complemented
160	C₂⁵	—	2	8	no	—
161	C₂⁵	—	2	8	no	—
162	C₂⁵	C₂×C₄×C₆₇	1	8	yes	—
163	C₂⁵	D₄×C₆₇	1	8	yes	—
164	C₂⁵	Q₈×C₆₇	1	8	yes	—
165	C₂⁵	C₂³×C₆₇	1	6	yes	—
166	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
167	C₂³×C₄	—	2	8	no	—
168	C₂³×C₄	—	2	128	no	—
169	C₂³×C₄	—	2	96	no	—
170	C₂³×C₄	—	2	16	no	—
171	C₂³×C₄	—	2	16	no	—
172	C₂³×C₄	—	2	16	no	—
173	C₂³×C₄	C₂×C₄×C₆₇	1	8	yes	—
174	C₂³×C₄	C₂×C₄×C₆₇	1	8	yes	—
175	C₂³×C₄	C₂×C₄×C₆₇	1	96	yes	complemented
176	C₂³×C₄	D₄×C₆₇	1	16	yes	—
177	C₂³×C₄	C₂³×C₆₇	1	24	yes	—
178	C₂³×C₄	C₂³×C₆₇	1	6	yes	—
179	C₂³×C₄	C₂³×C₆₇	1	16	yes	—
180	C₂³×C₄	C₂³×C₆₇	1	6	yes	—
181	C₂×C₄⋊C₄	—	2	96	no	—
182	C₂×C₄⋊C₄	C₂³×C₆₇	1	32	yes	complemented
183	C₂×C₂²⋊C₄	—	2	96	no	—
184	C₂×C₂²⋊C₄	C₂³×C₆₇	1	32	yes	—
185	#95	C₂³×C₆₇	1	32	yes	complemented
186	C₂⁴	—	2	32	no	—
187	C₂⁴	—	2	32	no	—
188	C₂⁴	—	2	64	no	—
189	C₂⁴	—	2	64	no	—
190	C₂⁴	—	2	24	no	—
191	C₂⁴	—	2	24	no	—
192	C₂⁴	C₄⋊C₄×C₆₇	1	64	yes	complemented
193	C₂⁴	C₂²×C₄×C₆₇	1	24	yes	—
194	C₂⁴	C₂²×C₄×C₆₇	1	4	yes	—
195	C₂⁴	C₂×D₄×C₆₇	1	24	yes	—
196	C₂⁴	C₂×D₄×C₆₇	1	8	yes	—
197	C₂⁴	C₂×D₄×C₆₇	1	4	yes	—
198	C₂⁴	C₂×Q₈×C₆₇	1	24	yes	—
199	C₂⁴	C₂×Q₈×C₆₇	1	4	yes	—
200	C₂⁴	#41×C₆₇	1	8	yes	—
201	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
202	C₂⁴	C₂⁴×C₆₇	1	3	yes	—
203	C₂²×C₄	—	2	32	no	—
204	C₂²×C₄	—	2	24	no	—
205	C₂²×C₄	—	2	384	no	—
206	C₂²×C₄	—	2	64	no	—
207	C₂²×C₄	—	2	48	no	—
208	C₂²×C₄	—	2	48	no	—
209	C₂²×C₄	—	2	48	no	—
210	C₂²×C₄	—	4	64	no	—
211	C₂²×C₄	C₂²⋊C₄×C₆₇	1	64	yes	complemented
212	C₂²×C₄	C₂²×C₄×C₆₇	1	24	yes	—
213	C₂²×C₄	C₂²×C₄×C₆₇	1	24	yes	—
214	C₂²×C₄	C₂²×C₄×C₆₇	1	64	yes	complemented
215	C₂²×C₄	C₂×D₄×C₆₇	1	48	yes	—
216	C₂²×C₄	C₂⁴×C₆₇	1	16	yes	—
217	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
218	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
219	C₄⋊C₄	—	2	64	no	—
220	C₂²⋊C₄	—	2	64	no	—
221	C₂³	—	2	48	no	—
222	C₂³	—	2	48	no	—
223	C₂³	—	2	96	no	—
224	C₂³	—	2	96	no	—
225	C₂³	—	2	8	no	—
226	C₂³	—	2	8	no	—
227	C₂³	—	2	256	no	—
228	C₂³	C₂×C₂²⋊C₄×C₆₇	1	16	yes	—
229	C₂³	C₂×C₄⋊C₄×C₆₇	1	16	yes	—
230	C₂³	C₂×C₄⋊C₄×C₆₇	1	96	yes	complemented
231	C₂³	#117×C₆₇	1	16	yes	—
232	C₂³	#121×C₆₇	1	32	yes	—
233	C₂³	C₂²⋊Q₈×C₆₇	1	32	yes	—
234	C₂³	C₂³×C₄×C₆₇	1	8	yes	—
235	C₂³	C₂³×C₄×C₆₇	1	6	yes	—
236	C₂³	C₂²×D₄×C₆₇	1	8	yes	—
237	C₂³	C₂²×D₄×C₆₇	1	12	yes	—
238	C₂³	C₂²×D₄×C₆₇	1	6	yes	—
239	C₂³	C₂²×Q₈×C₆₇	1	8	yes	—
240	C₂³	C₂²×Q₈×C₆₇	1	6	yes	—
241	C₂³	C₂×#41×C₆₇	1	12	yes	—
242	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
243	C₂×C₄	—	2	48	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	2	128	no	—
246	C₂×C₄	—	2	16	no	—
247	C₂×C₄	—	2	16	no	—
248	C₂×C₄	—	2	16	no	—
249	C₂×C₄	—	4	96	no	—
250	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	96	yes	complemented
251	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
252	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
253	C₂×C₄	C₂²×D₄×C₆₇	1	16	yes	—
254	C₂²	—	2	192	no	—
255	C₂²	—	2	8	no	—
256	C₂²	—	2	8	no	—
257	C₂²	—	2	16	no	—
258	C₂²	—	2	16	no	—
259	C₂²	#380×C₆₇	1	64	yes	direct product factor
260	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	12	yes	—
261	C₂²	C₂²×C₄⋊C₄×C₆₇	1	12	yes	—
262	C₂²	C₂²×C₄⋊C₄×C₆₇	1	16	yes	complemented
263	C₂²	C₂×#117×C₆₇	1	12	yes	—
264	C₂²	C₂×#121×C₆₇	1	24	yes	—
265	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	24	yes	—
266	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
267	C₂²	C₂³×D₄×C₆₇	1	2	yes	—
268	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
269	C₂²	C₂³×Q₈×C₆₇	1	1	yes	characteristic
270	C₂²	C₂²×#41×C₆₇	1	2	yes	—
271	C₄	—	2	8	no	—
272	C₄	—	4	16	no	—
273	C₄	C₂²×C₂²⋊C₄×C₆₇	1	16	yes	complemented
274	C₂	—	2	16	no	—
275	C₂	C₂×#380×C₆₇	1	24	yes	direct product factor
276	C₂	C₂³×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
277	C₂	C₂³×C₄⋊C₄×C₆₇	1	1	yes	characteristic
278	C₂	C₂²×#117×C₆₇	1	1	yes	characteristic
279	C₂	C₂²×#121×C₆₇	1	2	yes	—
280	C₂	C₂²×C₂²⋊Q₈×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁵×C₆₇	C₂³
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂⁶×C₆₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₄⋊C₄×C₆₇	C₂	semidirect
4. C₂×#380×C₆₇	C₂	direct (N × H)
5. C₂²×#95×C₆₇	C₂	semidirect
8. C₂⁴×C₄×C₆₇	C₄	semidirect
14. C₂²×C₄⋊C₄×C₆₇	C₂²	semidirect
17. #380×C₆₇	C₂²	direct (N × H)
18. C₂×#95×C₆₇	C₂²	semidirect
34. C₂³×C₄×C₆₇	C₂×C₄	semidirect
41. C₂×C₄⋊C₄×C₆₇	C₂³	semidirect
44. #95×C₆₇	C₂³	semidirect
51. C₂⁴×C₆₇	C₄⋊C₄	semidirect
70. C₂²×C₄×C₆₇	C₂²⋊C₄	semidirect
73. C₂²×C₄×C₆₇	C₂²×C₄	semidirect
89. C₂³×C₆₇	C₂×C₄⋊C₄	semidirect
109. C₂×C₄×C₆₇	C₂×C₂²⋊C₄	semidirect
118. C₂²×C₆₇	#380	direct (N × H)
121. C₂²×C₆₇	C₂²×C₄⋊C₄	semidirect
132. C₄×C₆₇	C₂²×C₂²⋊C₄	semidirect
133. C₂²×#380	C₆₇	direct (N × H)
135. C₂×C₆₇	C₂×#380	direct (N × H)
142. C₂³×C₄⋊C₄	C₂×C₆₇	semidirect
144. C₂×#380	C₂×C₆₇	direct (N × H)
145. C₂²×#95	C₂×C₆₇	semidirect
146. C₆₇	C₂²×#380	direct (N × H)
149. C₂⁴×C₄	C₄×C₆₇	semidirect
155. C₂²×C₄⋊C₄	C₂²×C₆₇	semidirect
158. #380	C₂²×C₆₇	direct (N × H)
159. C₂×#95	C₂²×C₆₇	semidirect
175. C₂³×C₄	C₂×C₄×C₆₇	semidirect
182. C₂×C₄⋊C₄	C₂³×C₆₇	semidirect
185. #95	C₂³×C₆₇	semidirect
192. C₂⁴	C₄⋊C₄×C₆₇	semidirect
211. C₂²×C₄	C₂²⋊C₄×C₆₇	semidirect
214. C₂²×C₄	C₂²×C₄×C₆₇	semidirect
230. C₂³	C₂×C₄⋊C₄×C₆₇	semidirect
250. C₂×C₄	C₂×C₂²⋊C₄×C₆₇	semidirect
259. C₂²	#380×C₆₇	direct (N × H)
262. C₂²	C₂²×C₄⋊C₄×C₆₇	semidirect
273. C₄	C₂²×C₂²⋊C₄×C₆₇	semidirect
275. C₂	C₂×#380×C₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×C₄×C₆₇	2
C₂³×C₄⋊C₄×C₆₇	2
C₂³×C₂²⋊C₄×C₆₇	2
C₂×#380×C₆₇	2
C₂²×#95×C₆₇	2
C₂²×#380	67

Subgroups — C22×#380×C67 #30713255

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#380×C₆₇ #30713255