Subgroups — #31009342

Order 17152.

530 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
2	#22562116	C₂	1	4	yes	maximal, complemented
3	#22562114	C₂	1	2	yes	maximal, complemented
4	#22562111	C₂	1	2	yes	maximal, complemented
5	#22561964	C₂	1	2	yes	maximal
6	C₄×#14523983	C₂	1	1	yes	maximal, characteristic
7	M₃₂×Dic₆₇	C₂	1	2	yes	maximal
8	C₆₇×#2150	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₃₄	C₄	1	1	yes	characteristic
10	C₂×#14523983	C₂²	1	1	yes	characteristic
11	C₂×#14523983	C₂²	1	1	yes	characteristic
12	C₈×Dic₁₃₄	C₄	1	2	yes	—
13	C₈×Dic₁₃₄	C₄	1	1	yes	characteristic
14	C₈×Dic₁₃₄	C₄	1	1	yes	characteristic
15	C₈×Dic₁₃₄	C₂²	1	1	yes	characteristic
16	C₈×Dic₁₃₄	C₂²	1	1	yes	characteristic
17	C₈×Dic₁₃₄	C₂²	1	2	yes	—
18	C₂×C₄⋊(C₆₇⋊C₈)	C₄	1	1	yes	characteristic
19	C₂×Dic₆₇⋊C₈	C₄	1	2	yes	—
20	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
21	Dic₆₇⋊C₁₆	C₂²	1	4	yes	—
22	Dic₆₇⋊C₁₆	C₂²	1	4	yes	—
23	#16863172	C₂²	1	2	yes	—
24	#16863171	C₂²	1	4	yes	—
25	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
26	C₁₆×Dic₆₇	C₂²	1	4	yes	—
27	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	—
30	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	—
31	C₂²×Dic₁₃₄	C₈	1	1	yes	characteristic
32	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
33	C₄×Dic₁₃₄	C₈	1	2	yes	—
34	C₄×Dic₁₃₄	C₈	1	1	yes	characteristic
35	C₄×Dic₁₃₄	C₈	1	1	yes	characteristic
36	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
37	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
38	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊Dic₆₇	C₈	1	1	yes	characteristic
40	C₂×Dic₆₇⋊C₄	C₈	1	2	yes	—
41	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
42	#14523983	—	2	2	no	—
43	#14523983	Q₈	1	4	yes	complemented
44	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
45	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
46	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
47	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
48	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
49	Dic₆₇⋊C₈	C₂×C₄	1	2	yes	—
50	Dic₆₇⋊C₈	C₂×C₄	1	2	yes	—
51	Dic₆₇⋊C₈	C₂×C₄	1	2	yes	—
52	Dic₆₇⋊C₈	C₂×C₄	1	2	yes	—
53	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
54	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
55	C₈×Dic₆₇	C₂³	1	2	yes	—
56	C₈×Dic₆₇	C₂³	1	2	yes	—
57	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
60	M₃₂×C₆₇	—	2	4	no	—
61	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
62	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
66	C₂×Dic₁₃₄	—	2	4	no	—
67	C₂×Dic₁₃₄	C₂×C₈	1	1	yes	characteristic
68	C₂×Dic₁₃₄	C₂×C₈	1	2	yes	—
69	C₂×Dic₁₃₄	C₂×C₈	1	1	yes	characteristic
70	C₂²×Dic₆₇	C₂×C₈	1	2	yes	—
71	C₂²×Dic₆₇	C₂×C₈	1	2	yes	—
72	C₂×C₆₇⋊C₈	—	2	4	no	—
73	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
74	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
75	C₄⋊Dic₆₇	C₂×C₈	1	2	yes	—
76	C₄⋊Dic₆₇	C₂×C₈	1	1	yes	characteristic
77	C₄⋊Dic₆₇	C₂×C₈	1	1	yes	characteristic
78	Dic₆₇⋊C₄	C₂×C₈	1	2	yes	—
79	Dic₆₇⋊C₄	C₂×C₈	1	2	yes	—
80	Dic₆₇⋊C₄	C₂×C₈	1	2	yes	—
81	Dic₆₇⋊C₄	C₂×C₈	1	2	yes	—
82	C₄×Dic₆₇	C₂×C₈	1	2	yes	—
83	C₄×Dic₆₇	C₂×C₈	1	2	yes	—
84	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
85	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
86	C₆₇⋊C₁₆	—	2	2	no	—
87	C₆₇⋊C₁₆	C₂×Q₈	1	4	yes	complemented
88	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₆₇	—	2	2	no	—
92	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
96	C₂×C₈×C₆₇	#41	1	2	yes	—
97	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
98	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
99	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
100	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
102	C₁₆×C₆₇	—	2	4	no	—
103	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₆₇	—	2	2	no	—
105	C₂×C₄×C₆₇	—	4	1	no	—
106	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
107	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₆₇	#131	1	2	yes	—
113	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
114	C₂×Dic₆₇	—	2	4	no	—
115	C₂×Dic₆₇	—	4	2	no	—
116	C₂×Dic₆₇	C₂²×C₈	1	2	yes	—
117	C₂×Dic₆₇	C₂²×C₈	1	2	yes	—
118	C₂×Dic₆₇	C₂²×C₈	1	2	yes	—
119	C₂×Dic₆₇	C₂²×C₈	1	2	yes	—
120	Dic₁₃₄	—	2	2	no	—
121	Dic₁₃₄	—	2	4	no	—
122	Dic₁₃₄	M₃₂	1	4	yes	complemented
123	C₈×C₆₇	—	2	1	no	—
124	C₈×C₆₇	—	2	1	no	—
125	C₈×C₆₇	C₄×Q₈	1	2	yes	—
126	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
127	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
128	C₆₇⋊C₈	—	2	2	no	—
129	C₆₇⋊C₈	—	2	2	no	—
130	C₂²×C₆₇	—	2	1	no	—
131	C₂²×C₆₇	—	2	1	no	—
132	C₂²×C₆₇	C₈×Q₈	1	2	yes	—
133	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
134	C₄×C₆₇	—	2	1	no	—
135	C₄×C₆₇	—	2	1	no	—
136	C₄×C₆₇	—	2	1	no	—
137	C₄×C₆₇	C₈×Q₈	1	2	yes	—
138	C₄×C₆₇	C₂×M₃₂	1	2	yes	—
139	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
140	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
141	Dic₆₇	—	2	2	no	—
142	Dic₆₇	—	2	2	no	—
143	Dic₆₇	—	2	2	no	—
144	Dic₆₇	C₂×M₃₂	1	4	yes	—
145	Q₈×M₃₂	—	67	1	no	maximal, Sylow
146	C₂×C₆₇	—	2	2	no	—
147	C₂×C₆₇	C₂×C₈×Q₈	1	1	yes	characteristic
148	C₂×C₆₇	C₂²×M₃₂	1	1	yes	characteristic
149	C₂×C₆₇	#3407	1	1	yes	characteristic
150	C₂×C₈×Q₈	—	67	1	no	—
151	#2183	—	67	2	no	—
152	#2183	—	67	4	no	—
153	Q₈×C₁₆	—	67	2	no	—
154	#2150	—	67	2	no	—
155	#2150	D₆₇	1	1	yes	characteristic
156	C₄×M₃₂	—	67	2	no	—
157	C₄×M₃₂	—	67	1	no	—
158	C₆₇	Q₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
159	C₂×C₄×Q₈	—	67	1	no	—
160	C₂×M₃₂	—	67	1	no	—
161	C₂×M₃₂	—	67	1	no	—
162	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
163	C₈×Q₈	—	67	2	no	—
164	C₈×Q₈	—	67	1	no	—
165	C₈×Q₈	—	67	1	no	—
166	C₈×Q₈	—	67	1	no	—
167	C₈×Q₈	—	67	2	no	—
168	C₈×Q₈	—	67	1	no	—
169	C₂×C₄⋊C₈	—	67	2	no	—
170	C₂×C₄⋊C₈	—	67	1	no	—
171	C₂×C₄×C₈	—	67	2	no	—
172	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
173	C₄⋊C₁₆	—	67	4	no	—
174	C₄⋊C₁₆	—	67	4	no	—
175	C₄⋊C₁₆	C₂×D₆₇	1	2	yes	—
176	C₄⋊C₁₆	C₂×D₆₇	1	2	yes	—
177	#346	—	67	4	no	—
178	#346	—	67	2	no	—
179	C₄×C₁₆	—	67	4	no	—
180	C₄×C₁₆	—	67	2	no	—
181	C₂²×Q₈	—	67	1	no	—
182	C₂²×C₈	—	67	2	no	—
183	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
185	C₄×Q₈	—	67	1	no	—
186	C₄×Q₈	—	67	2	no	—
187	C₄×Q₈	—	67	1	no	—
188	C₄×Q₈	—	67	1	no	—
189	C₄×Q₈	—	67	2	no	—
190	C₄×Q₈	—	67	1	no	—
191	C₂×C₄⋊C₄	—	67	2	no	—
192	C₂×C₄⋊C₄	—	67	1	no	—
193	C₂×C₄²	—	67	2	no	—
194	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
195	M₃₂	—	2	4	no	—
196	M₃₂	—	67	4	no	—
197	M₃₂	—	134	2	no	—
198	C₂×C₁₆	—	67	2	no	—
199	C₂×C₁₆	—	67	2	no	—
200	C₂×C₁₆	C₂²×D₆₇	1	4	yes	—
201	C₄⋊C₈	—	67	2	no	—
202	C₄⋊C₈	—	67	2	no	—
203	C₄⋊C₈	—	67	2	no	—
204	C₄⋊C₈	—	67	2	no	—
205	C₄⋊C₈	—	67	1	no	—
206	C₄⋊C₈	—	67	2	no	—
207	C₄⋊C₈	—	67	1	no	—
208	C₄×C₈	—	67	2	no	—
209	C₄×C₈	—	67	2	no	—
210	C₄×C₈	—	67	2	no	—
211	C₄×C₈	—	67	2	no	—
212	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
213	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
214	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
215	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
216	C₂×Q₈	—	67	1	no	—
217	C₂×Q₈	—	67	2	no	—
218	C₂×Q₈	—	67	1	no	—
219	C₂×Q₈	—	134	4	no	—
220	C₂²×C₄	—	67	2	no	—
221	C₂²×C₄	—	67	2	no	—
222	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
223	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	67	2	no	—
227	C₂×C₈	—	67	2	no	—
228	C₂×C₈	—	134	4	no	—
229	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
230	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
231	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
232	C₂×C₈	Q₈×D₆₇	1	2	yes	—
233	C₂×C₈	#11761627	1	2	yes	—
234	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
235	C₄⋊C₄	—	67	2	no	—
236	C₄⋊C₄	—	67	2	no	—
237	C₄⋊C₄	—	67	2	no	—
238	C₄⋊C₄	—	67	2	no	—
239	C₄⋊C₄	—	67	2	no	—
240	C₄⋊C₄	—	67	1	no	—
241	C₄⋊C₄	—	67	1	no	—
242	C₄²	—	67	2	no	—
243	C₄²	—	67	2	no	—
244	C₄²	—	67	2	no	—
245	C₄²	—	67	2	no	—
246	C₄²	C₈×D₆₇	1	1	yes	characteristic
247	C₄²	C₈×D₆₇	1	1	yes	characteristic
248	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
249	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
250	C₁₆	—	2	4	no	—
251	C₁₆	—	67	4	no	—
252	C₁₆	—	134	2	no	—
253	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
254	Q₈	—	67	4	no	—
255	Q₈	—	134	4	no	—
256	Q₈	—	134	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	268	2	no	—
260	C₂×C₄	—	67	2	no	—
261	C₂×C₄	—	67	2	no	—
262	C₂×C₄	—	67	2	no	—
263	C₂×C₄	—	67	2	no	—
264	C₂×C₄	—	134	4	no	—
265	C₂×C₄	#14524008	1	2	yes	—
266	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
267	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
268	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
269	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
270	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
271	C₂×C₄	#14524051	1	2	yes	—
272	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
273	C₈	—	2	1	no	—
274	C₈	—	2	1	no	—
275	C₈	—	134	2	no	—
276	C₈	—	134	2	no	—
277	C₈	#14524008	1	2	yes	—
278	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
279	C₈	C₂×#11761627	1	1	yes	characteristic
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	#16863520	1	2	yes	—
283	C₂²	C₂²×C₈×D₆₇	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	67	4	no	—
288	C₄	—	134	2	no	—
289	C₄	—	134	2	no	—
290	C₄	—	134	2	no	—
291	C₄	#16863520	1	2	yes	—
292	C₄	M₃₂×D₆₇	1	2	yes	—
293	C₄	C₂×#14524008	1	1	yes	characteristic
294	C₄	C₂×#14524051	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	C₂×#16863520	1	1	yes	characteristic
297	C₂	C₂×M₃₂×D₆₇	1	1	yes	characteristic
298	C₂	#22573311	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₈
Fitting	C₆₇×#2150	C₂
prime radical	C₂³×C₆₇	C₂²×C₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	#2150	D₆₇
hypocenter	C₆₇	Q₈×M₃₂
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
2. #22562116	C₂	semidirect
3. #22562114	C₂	semidirect
4. #22562111	C₂	semidirect
43. #14523983	Q₈	semidirect
87. C₆₇⋊C₁₆	C₂×Q₈	semidirect
122. Dic₁₃₄	M₃₂	semidirect
158. C₆₇	Q₈×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₃₄	2
#22562116	2
#22562114	2
#22562111	2
#22561964	2
C₄×#14523983	2
M₃₂×Dic₆₇	2
C₆₇×#2150	2
Q₈×M₃₂	67