Subgroups — #30853894

Order 17152.

1142 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863434	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14523953	C₂	1	1	yes	maximal, characteristic
3	#117×Dic₆₇	C₂	1	2	yes	maximal
4	#22560109	C₂	1	4	yes	maximal, complemented
5	#22560108	C₂	1	4	yes	maximal, complemented
6	#22559925	C₂	1	2	yes	maximal
7	C₂×#427×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×#14524002	C₂²	1	2	yes	—
10	C₂²×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
11	#16863434	C₄	1	8	yes	complemented
12	C₂×#14523977	C₂²	1	2	yes	—
13	C₂×#14523975	C₂²	1	2	yes	—
14	C₂×#14523953	C₄	1	8	yes	complemented
15	C₂×#14523953	C₂²	1	2	yes	—
16	C₂×#14523953	C₂²	1	2	yes	—
17	C₄⋊C₄×Dic₆₇	C₂²	1	8	yes	—
18	C₂²⋊C₄×Dic₆₇	—	2	4	no	—
19	#16863242	—	2	2	no	—
20	C₄²×Dic₆₇	—	2	2	no	—
21	#16863158	C₂²	1	8	yes	—
22	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#117×C₆₇	C₂²	1	2	yes	—
24	#427×C₆₇	C₄	1	8	yes	complemented
25	C₂×#107×C₆₇	C₂²	1	2	yes	—
26	C₂×#106×C₆₇	C₂²	1	2	yes	—
27	C₂³×Dic₆₇	D₄	1	2	yes	—
28	C₂²×C₆₇⋊C₈	—	2	2	no	—
29	C₂²×C₆₇⋊C₈	—	2	2	no	—
30	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
31	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
32	#14524002	—	2	4	no	—
33	#14524002	C₂×C₄	1	8	yes	—
34	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
35	C₂×C₂²⋊Dic₆₇	—	2	2	no	—
36	C₂×C₄×Dic₆₇	—	2	8	no	—
37	C₂×C₄×Dic₆₇	—	2	2	no	—
38	C₂×C₄×Dic₆₇	C₂×C₄	1	4	yes	—
39	C₂×C₄×Dic₆₇	D₄	1	2	yes	—
40	C₂×C₄×Dic₆₇	Q₈	1	4	yes	—
41	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
42	#14523977	C₂×C₄	1	8	yes	complemented
43	#14523975	C₂×C₄	1	8	yes	complemented
44	#14523953	C₂×C₄	1	8	yes	—
45	#14523953	C₂×C₄	1	8	yes	—
46	C₆₇⋊#95	—	2	8	no	—
47	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
50	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
53	#117×C₆₇	—	2	4	no	—
54	#117×C₆₇	C₂×C₄	1	8	yes	—
55	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
56	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
57	C₂×C₄²×C₆₇	—	2	2	no	—
58	#107×C₆₇	C₂×C₄	1	8	yes	complemented
59	#106×C₆₇	C₂×C₄	1	8	yes	complemented
60	C₂²×Dic₆₇	—	2	4	no	—
61	C₂²×Dic₆₇	—	4	4	no	—
62	C₂²×Dic₆₇	C₄⋊C₄	1	8	yes	—
63	C₂²×Dic₆₇	C₂×D₄	1	4	yes	—
64	C₂×C₆₇⋊C₈	—	2	8	no	—
65	C₂×C₆₇⋊C₈	—	2	2	no	—
66	C₂×C₆₇⋊C₈	—	2	2	no	—
67	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
68	C₄⋊Dic₆₇	—	2	4	no	—
69	C₄⋊Dic₆₇	C₂²×C₄	1	8	yes	—
70	C₂²⋊Dic₆₇	—	2	8	no	—
71	C₄×Dic₆₇	—	2	8	no	—
72	C₄×Dic₆₇	—	4	16	no	—
73	C₄×Dic₆₇	C₄⋊C₄	1	8	yes	—
74	C₄×Dic₆₇	C₂²×C₄	1	4	yes	—
75	C₄×Dic₆₇	C₂×Q₈	1	4	yes	—
76	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	—	2	4	no	—
78	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
83	C₂²×C₄×C₆₇	#41	1	2	yes	—
84	C₂²×C₄×C₆₇	#41	1	2	yes	—
85	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
86	C₂×C₈×C₆₇	C₄²	1	16	yes	complemented
87	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
93	C₄⋊C₄×C₆₇	—	2	4	no	—
94	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
95	C₂²⋊C₄×C₆₇	—	2	8	no	—
96	C₄²×C₆₇	—	2	8	no	—
97	C₂³×C₆₇	—	2	4	no	—
98	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
99	C₂³×C₆₇	C₄×D₄	1	2	yes	—
100	C₂³×C₆₇	C₄×D₄	1	2	yes	—
101	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₆₇	—	2	8	no	—
103	C₂×C₄×C₆₇	—	4	8	no	—
104	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
105	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₆₇	#117	1	4	yes	—
107	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
110	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
111	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
117	C₂×Dic₆₇	—	2	8	no	—
118	C₂×Dic₆₇	—	2	8	no	—
119	C₂×Dic₆₇	—	4	8	no	—
120	C₂×Dic₆₇	—	4	8	no	—
121	C₂×Dic₆₇	C₂×C₄⋊C₄	1	8	yes	—
122	C₈×C₆₇	C₂×C₄²	1	4	yes	—
123	C₈×C₆₇	C₂×C₄²	1	4	yes	—
124	C₆₇⋊C₈	—	2	4	no	—
125	C₂²×C₆₇	—	2	4	no	—
126	C₂²×C₆₇	—	2	4	no	—
127	C₂²×C₆₇	—	2	4	no	—
128	C₂²×C₆₇	C₄×C₄⋊C₄	1	4	yes	—
129	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
130	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
131	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂²×C₆₇	#575	1	4	yes	—
133	C₄×C₆₇	—	4	8	no	—
134	C₄×C₆₇	C₄×C₄⋊C₄	1	4	yes	—
135	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
136	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
137	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
138	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
139	Dic₆₇	—	2	8	no	—
140	Dic₆₇	—	4	8	no	—
141	#20451	—	67	1	no	maximal, Sylow
142	C₂×C₆₇	—	2	4	no	—
143	C₂×C₆₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₆₇	#2911	1	2	yes	—
145	C₂×C₆₇	#2943	1	2	yes	—
146	C₂×C₆₇	#2943	1	2	yes	—
147	C₂×#427	—	67	1	no	—
148	C₂×#427	D₆₇	1	1	yes	characteristic
149	C₂²×#97	—	67	1	no	—
150	C₄×#117	—	67	2	no	—
151	#1776	—	67	4	no	—
152	#1776	—	67	4	no	—
153	#1735	—	67	2	no	—
154	C₆₇	#20451	1	1	yes	characteristic, complemented, Sylow
155	C₂³×C₈	—	67	1	no	—
156	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
157	C₂×#117	—	67	2	no	—
158	C₂×#117	C₂×D₆₇	1	2	yes	—
159	C₂²×C₄²	—	67	1	no	—
160	#427	—	67	8	no	—
161	#427	Dic₆₇	1	8	yes	complemented
162	C₂×#107	—	67	2	no	—
163	C₂×#107	C₂×D₆₇	1	2	yes	—
164	C₂×#106	—	67	2	no	—
165	C₂×#106	C₂×D₆₇	1	2	yes	—
166	C₂×#97	—	67	8	no	—
167	C₂×#97	—	67	2	no	—
168	C₂×#97	—	67	2	no	—
169	C₄×C₄⋊C₄	—	67	8	no	—
170	C₄×C₂²⋊C₄	—	134	4	no	—
171	#376	—	134	2	no	—
172	C₄³	—	134	2	no	—
173	#340	—	67	8	no	—
174	C₂³×C₄	—	67	2	no	—
175	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
176	C₂²×C₈	—	67	4	no	—
177	C₂²×C₈	—	67	2	no	—
178	C₂²×C₈	—	134	2	no	—
179	C₂²×C₈	—	134	2	no	—
180	C₂²×C₈	C₄×D₆₇	1	4	yes	—
181	C₂²×C₈	C₄×D₆₇	1	4	yes	—
182	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
183	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
185	#117	—	2	4	no	—
186	#117	—	67	8	no	—
187	#117	—	134	4	no	—
188	#117	C₂×Dic₆₇	1	8	yes	—
189	C₂×C₄⋊C₄	—	67	4	no	—
190	C₂×C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
191	C₂×C₂²⋊C₄	—	2	2	no	—
192	C₂×C₂²⋊C₄	—	134	2	no	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	—	67	4	no	—
195	C₂×C₄²	—	67	2	no	—
196	C₂×C₄²	—	67	4	no	—
197	C₂×C₄²	—	67	2	no	—
198	C₂×C₄²	—	134	8	no	—
199	C₂×C₄²	—	134	2	no	—
200	#107	—	67	8	no	—
201	#107	C₂×Dic₆₇	1	8	yes	complemented
202	#106	—	67	8	no	—
203	#106	C₂×Dic₆₇	1	8	yes	complemented
204	#97	—	67	8	no	—
205	#97	—	67	8	no	—
206	#95	—	134	8	no	—
207	C₂⁴	D₄×D₆₇	1	1	yes	characteristic
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	268	4	no	—
210	C₂²×C₄	—	67	8	no	—
211	C₂²×C₄	—	67	4	no	—
212	C₂²×C₄	—	134	4	no	—
213	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
214	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
215	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
216	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
217	C₂²×C₄	Q₈×D₆₇	1	2	yes	—
218	C₂²×C₄	#11761627	1	2	yes	—
219	C₂²×C₄	#11761629	1	2	yes	—
220	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
221	C₂×C₈	—	67	4	no	—
222	C₂×C₈	—	134	8	no	—
223	C₂×C₈	—	134	2	no	—
224	C₂×C₈	—	134	2	no	—
225	C₂×C₈	C₄×Dic₆₇	1	16	yes	complemented
226	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
227	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
228	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
229	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
230	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
231	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	67	8	no	—
234	C₄⋊C₄	—	134	4	no	—
235	C₄⋊C₄	C₂²×Dic₆₇	1	8	yes	—
236	C₂²⋊C₄	—	2	8	no	—
237	C₂²⋊C₄	—	134	8	no	—
238	C₄²	—	2	8	no	—
239	C₄²	—	268	16	no	—
240	C₄²	—	67	8	no	—
241	C₄²	—	67	4	no	—
242	C₄²	—	67	4	no	—
243	C₄²	—	134	8	no	—
244	C₂³	—	2	4	no	—
245	C₂³	C₄⋊C₄×D₆₇	1	2	yes	—
246	C₂³	#14524005	1	2	yes	—
247	C₂³	D₄×Dic₆₇	1	2	yes	—
248	C₂³	C₂×D₄×D₆₇	1	1	yes	characteristic
249	C₂×C₄	—	2	8	no	—
250	C₂×C₄	—	4	8	no	—
251	C₂×C₄	—	268	8	no	—
252	C₂×C₄	—	268	8	no	—
253	C₂×C₄	—	67	8	no	—
254	C₂×C₄	—	134	8	no	—
255	C₂×C₄	—	134	8	no	—
256	C₂×C₄	C₂×C₄×Dic₆₇	1	4	yes	—
257	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
258	C₂×C₄	#14524001	1	4	yes	—
259	C₂×C₄	#14524005	1	2	yes	—
260	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
261	C₂×C₄	#14524008	1	4	yes	—
262	C₂×C₄	Q₈×Dic₆₇	1	4	yes	—
263	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
264	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
265	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
266	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
267	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
268	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
269	C₈	—	134	4	no	—
270	C₈	C₂×C₄×Dic₆₇	1	4	yes	—
271	C₈	C₂×C₄×Dic₆₇	1	4	yes	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	C₄⋊C₄×Dic₆₇	1	4	yes	—
276	C₂²	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
277	C₂²	C₂×#14524005	1	1	yes	characteristic
278	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
279	C₂²	#16864193	1	4	yes	—
280	C₄	—	4	8	no	—
281	C₄	—	268	8	no	—
282	C₄	—	134	8	no	—
283	C₄	C₄⋊C₄×Dic₆₇	1	4	yes	—
284	C₄	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
285	C₄	C₂×#14524001	1	1	yes	characteristic
286	C₄	C₂×#14524008	1	1	yes	characteristic
287	C₄	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
288	C₂	—	2	4	no	—
289	C₂	C₂×C₄⋊C₄×Dic₆₇	1	1	yes	characteristic
290	C₂	#22568797	1	2	yes	—
291	C₂	#22569009	1	2	yes	—
292	C₂	#22569010	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₇
commutator	C₄×C₆₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#427×C₆₇	C₂
socle	C₂³×C₆₇	C₂²×D₄
prime radical	C₂⁴×C₆₇	C₂×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#427	D₆₇
hypocenter	C₆₇	#20451
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
4. #22560109	C₂	semidirect
5. #22560108	C₂	semidirect
11. #16863434	C₄	semidirect
14. C₂×#14523953	C₄	semidirect
24. #427×C₆₇	C₄	semidirect
42. #14523977	C₂×C₄	semidirect
43. #14523975	C₂×C₄	semidirect
58. #107×C₆₇	C₂×C₄	semidirect
59. #106×C₆₇	C₂×C₄	semidirect
86. C₂×C₈×C₆₇	C₄²	semidirect
154. C₆₇	#20451	semidirect
161. #427	Dic₆₇	semidirect
201. #107	C₂×Dic₆₇	semidirect
203. #106	C₂×Dic₆₇	semidirect
225. C₂×C₈	C₄×Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863434	2
C₂²×#14523953	2
#117×Dic₆₇	2
#22560109	2
#22560108	2
#22559925	2
C₂×#427×C₆₇	2
#20451	67