Subgroups — #30770923

Order 17152.

474 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523983	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863172	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#22558287	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558287	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#14523983	—	2	2	no	—
9	C₂×#14523983	—	2	2	no	—
10	C₂×#14523983	C₄	1	4	yes	complemented
11	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂²⋊(C₆₇⋊C₁₆)	C₄	1	4	yes	complemented
16	#16863172	—	2	2	no	—
17	C₄×C₆₇⋊C₁₆	—	2	2	no	—
18	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₈×C₆₇	C₄	1	1	yes	characteristic
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₄×C₂²⋊C₄×C₆₇	C₄	1	1	yes	characteristic
23	#336×C₆₇	C₄	1	1	yes	characteristic
24	#336×C₆₇	C₄	1	1	yes	characteristic
25	#14523983	—	2	8	no	—
26	C₂×C₆₇⋊C₁₆	—	2	1	no	—
27	C₂×C₆₇⋊C₁₆	—	2	1	no	—
28	C₂×C₆₇⋊C₁₆	—	2	1	no	—
29	C₂×C₆₇⋊C₁₆	—	2	1	no	—
30	C₂×C₆₇⋊C₁₆	—	2	1	no	—
31	C₂×C₆₇⋊C₁₆	—	2	1	no	—
32	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
33	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
34	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
35	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
36	C₂×C₆₇⋊C₁₆	D₄	1	2	yes	—
37	C₂×C₆₇⋊C₁₆	D₄	1	2	yes	—
38	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	—	2	1	no	—
40	C₂²×C₈×C₆₇	—	2	1	no	—
41	C₂²×C₈×C₆₇	—	2	1	no	—
42	C₂²×C₈×C₆₇	—	2	1	no	—
43	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
49	C₂×C₂²⋊C₄×C₆₇	C₈	1	2	yes	—
50	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂²⋊C₈×C₆₇	C₈	1	4	yes	—
53	C₄×C₈×C₆₇	—	2	1	no	—
54	C₄×C₈×C₆₇	—	2	1	no	—
55	C₄×C₈×C₆₇	—	2	1	no	—
56	C₄×C₈×C₆₇	—	2	1	no	—
57	#95×C₆₇	C₈	1	2	yes	—
58	C₆₇⋊C₁₆	—	2	2	no	—
59	C₆₇⋊C₁₆	—	2	2	no	—
60	C₆₇⋊C₁₆	—	2	2	no	—
61	C₆₇⋊C₁₆	C₂²⋊C₄	1	4	yes	complemented
62	C₂⁴×C₆₇	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	—	2	1	no	—
64	C₂²×C₄×C₆₇	—	2	1	no	—
65	C₂²×C₄×C₆₇	—	2	1	no	—
66	C₂²×C₄×C₆₇	—	2	1	no	—
67	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
68	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
69	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
71	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
72	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₆₇	—	2	1	no	—
74	C₂×C₈×C₆₇	—	2	1	no	—
75	C₂×C₈×C₆₇	—	2	1	no	—
76	C₂×C₈×C₆₇	—	2	1	no	—
77	C₂×C₈×C₆₇	—	2	1	no	—
78	C₂×C₈×C₆₇	—	2	1	no	—
79	C₂×C₈×C₆₇	—	2	2	no	—
80	C₂×C₈×C₆₇	—	2	2	no	—
81	C₂×C₈×C₆₇	—	2	2	no	—
82	C₂×C₈×C₆₇	—	2	2	no	—
83	C₂×C₈×C₆₇	C₄²	1	2	yes	—
84	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
85	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
86	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
91	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
92	C₂²⋊C₄×C₆₇	—	2	4	no	—
93	C₄²×C₆₇	—	2	1	no	—
94	C₄²×C₆₇	—	2	1	no	—
95	C₄²×C₆₇	—	2	1	no	—
96	C₄²×C₆₇	—	2	1	no	—
97	C₂³×C₆₇	—	2	1	no	—
98	C₂³×C₆₇	—	2	1	no	—
99	C₂³×C₆₇	—	2	2	no	—
100	C₂³×C₆₇	—	2	2	no	—
101	C₂³×C₆₇	C₄×C₈	1	2	yes	—
102	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₆₇	—	2	2	no	—
104	C₂×C₄×C₆₇	—	2	1	no	—
105	C₂×C₄×C₆₇	—	2	1	no	—
106	C₂×C₄×C₆₇	—	2	2	no	—
107	C₂×C₄×C₆₇	—	2	2	no	—
108	C₂×C₄×C₆₇	—	2	2	no	—
109	C₂×C₄×C₆₇	—	2	2	no	—
110	C₂×C₄×C₆₇	—	2	2	no	—
111	C₂×C₄×C₆₇	—	2	2	no	—
112	C₂×C₄×C₆₇	—	2	2	no	—
113	C₂×C₄×C₆₇	—	2	2	no	—
114	C₂×C₄×C₆₇	C₄×C₈	1	2	yes	—
115	C₂×C₄×C₆₇	M₃₂	1	4	yes	complemented
116	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
121	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
122	C₈×C₆₇	—	2	2	no	—
123	C₈×C₆₇	—	2	2	no	—
124	C₈×C₆₇	—	2	2	no	—
125	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₈×C₆₇	#117	1	1	yes	characteristic
127	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
128	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
129	C₂²×C₆₇	—	2	2	no	—
130	C₂²×C₆₇	—	2	2	no	—
131	C₂²×C₆₇	—	2	2	no	—
132	C₂²×C₆₇	—	4	2	no	—
133	C₂²×C₆₇	—	4	2	no	—
134	C₂²×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
135	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
136	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
137	C₂²×C₆₇	C₂×M₃₂	1	2	yes	—
138	C₂²×C₆₇	#504	1	2	yes	—
139	C₄×C₆₇	—	2	4	no	—
140	C₄×C₆₇	—	2	2	no	—
141	C₄×C₆₇	—	4	2	no	—
142	C₄×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
143	C₄×C₆₇	#405	1	1	yes	characteristic
144	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
145	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
146	#12612	—	67	1	no	maximal, Sylow
147	C₂×C₆₇	—	4	2	no	—
148	C₂×C₆₇	C₈×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₆₇	C₄×M₃₂	1	1	yes	characteristic
150	C₂×C₆₇	#2108	1	1	yes	characteristic
151	C₂×C₆₇	#2168	1	2	yes	—
152	C₂×C₆₇	#2169	1	2	yes	—
153	C₂²×M₃₂	—	67	1	no	—
154	C₂×C₂²⋊C₁₆	—	67	1	no	—
155	C₂×#346	—	67	1	no	—
156	C₂×C₄×C₁₆	—	67	1	no	—
157	C₈×C₂²⋊C₄	D₆₇	1	1	yes	characteristic
158	#1374	—	67	1	no	—
159	#1374	—	67	1	no	—
160	C₆₇	#12612	1	1	yes	characteristic, complemented, Sylow
161	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
162	C₂×M₃₂	—	67	4	no	—
163	C₂×M₃₂	—	134	2	no	—
164	C₂×M₃₂	—	134	2	no	—
165	C₂²×C₁₆	—	67	1	no	—
166	C₂²×C₁₆	—	67	1	no	—
167	C₂²×C₁₆	—	67	1	no	—
168	C₂²×C₁₆	—	67	1	no	—
169	C₂×C₂²⋊C₈	Dic₆₇	1	1	yes	characteristic
170	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
171	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
172	C₄×C₂²⋊C₄	Dic₆₇	1	1	yes	characteristic
173	C₂²⋊C₁₆	—	67	4	no	—
174	#346	—	134	2	no	—
175	C₄×C₁₆	—	134	2	no	—
176	#336	Dic₆₇	1	1	yes	characteristic
177	#336	Dic₆₇	1	1	yes	characteristic
178	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	—	2	1	no	—
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	2	1	no	—
183	C₂²×C₈	C₄×D₆₇	1	2	yes	—
184	C₂²×C₈	C₄×D₆₇	1	2	yes	—
185	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
186	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
187	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
189	C₂×C₂²⋊C₄	C₆₇⋊C₈	1	2	yes	—
190	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
191	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
192	M₃₂	—	134	8	no	—
193	C₂×C₁₆	—	67	2	no	—
194	C₂×C₁₆	—	67	2	no	—
195	C₂×C₁₆	—	67	2	no	—
196	C₂×C₁₆	—	67	2	no	—
197	C₂×C₁₆	—	67	2	no	—
198	C₂×C₁₆	—	67	2	no	—
199	C₂×C₁₆	—	134	1	no	—
200	C₂×C₁₆	—	134	1	no	—
201	C₂×C₁₆	—	134	1	no	—
202	C₂×C₁₆	—	134	1	no	—
203	C₂×C₁₆	—	134	1	no	—
204	C₂×C₁₆	—	134	1	no	—
205	C₂²⋊C₈	C₆₇⋊C₈	1	4	yes	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	2	1	no	—
208	C₄×C₈	—	2	1	no	—
209	C₄×C₈	—	2	1	no	—
210	#95	C₆₇⋊C₈	1	2	yes	—
211	C₂⁴	C₂×C₆₇⋊C₈	1	1	yes	characteristic
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
217	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
218	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
220	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
221	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
233	C₂×C₈	C₂×C₆₇⋊C₈	1	2	yes	—
234	C₂×C₈	C₂×C₆₇⋊C₈	1	2	yes	—
235	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
236	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
237	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
238	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
239	C₂×C₈	#11761627	1	1	yes	characteristic
240	C₂×C₈	#11761627	1	1	yes	characteristic
241	C₂²⋊C₄	—	2	4	no	—
242	C₄²	—	2	1	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	2	1	no	—
246	C₁₆	—	67	4	no	—
247	C₁₆	—	134	2	no	—
248	C₁₆	—	134	2	no	—
249	C₁₆	—	134	2	no	—
250	C₂³	—	2	2	no	—
251	C₂³	—	2	2	no	—
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	C₄×C₆₇⋊C₈	1	2	yes	—
255	C₂³	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	C₄×C₆₇⋊C₈	1	2	yes	—
268	C₂×C₄	#14523983	1	4	yes	complemented
269	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
270	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
271	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
273	C₂×C₄	#14524050	1	1	yes	characteristic
274	C₂×C₄	#14524050	1	1	yes	characteristic
275	C₈	—	2	2	no	—
276	C₈	—	2	2	no	—
277	C₈	—	2	2	no	—
278	C₈	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
279	C₈	#14523999	1	1	yes	characteristic
280	C₈	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
281	C₈	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	—	4	2	no	—
287	C₂²	C₂×C₄×C₆₇⋊C₈	1	1	yes	characteristic
288	C₂²	D₄×C₆₇⋊C₈	1	1	yes	characteristic
289	C₂²	D₄×C₆₇⋊C₈	1	1	yes	characteristic
290	C₂²	C₂×#14523983	1	2	yes	—
291	C₂²	#16863813	1	2	yes	—
292	C₄	—	2	2	no	—
293	C₄	—	2	4	no	—
294	C₄	—	4	2	no	—
295	C₄	C₂²⋊C₄×Dic₆₇	1	1	yes	characteristic
296	C₄	#16863352	1	1	yes	characteristic
297	C₄	D₄×C₆₇⋊C₈	1	1	yes	characteristic
298	C₄	D₄×C₆₇⋊C₈	1	1	yes	characteristic
299	C₂	—	4	2	no	—
300	C₂	C₂²⋊C₄×C₆₇⋊C₈	1	1	yes	characteristic
301	C₂	C₄×#14523983	1	1	yes	characteristic
302	C₂	#22561806	1	1	yes	characteristic
303	C₂	#22562035	1	2	yes	—
304	C₂	#22562041	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂×C₄×C₈
Fitting	C₈×C₂²⋊C₄×C₆₇	C₂
socle	C₂³×C₆₇	C₂²×C₈
prime radical	C₂⁴×C₆₇	C₂×C₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₈×C₂²⋊C₄	D₆₇
hypocenter	C₆₇	#12612
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₈×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
3. C₂×#16863172	C₂	semidirect
4. C₂×C₄×C₆₇⋊C₁₆	C₂	semidirect
5. #22558287	C₂	semidirect
6. #22558287	C₂	semidirect
10. C₂×#14523983	C₄	semidirect
15. C₂²⋊(C₆₇⋊C₁₆)	C₄	semidirect
61. C₆₇⋊C₁₆	C₂²⋊C₄	semidirect
115. C₂×C₄×C₆₇	M₃₂	semidirect
160. C₆₇	#12612	semidirect
268. C₂×C₄	#14523983	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14523983	2
C₂×C₂²⋊(C₆₇⋊C₁₆)	2
C₂×#16863172	2
C₂×C₄×C₆₇⋊C₁₆	2
#22558287	2
C₈×C₂²⋊C₄×C₆₇	2
#12612	67