Subgroups — #30856009

Order 17152.

772 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863445	C₂	1	1	yes	maximal, characteristic
2	#22568627	C₂	1	1	yes	maximal, characteristic
3	#22560332	C₂	1	8	yes	maximal, complemented
4	C₂×#16863166	C₂	1	2	yes	maximal
5	C₂×#16863152	C₂	1	2	yes	maximal
6	C₆₇×#2911	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11761612	C₂²	1	2	yes	—
8	C₂²×#11761612	C₂²	1	1	yes	characteristic
9	C₂×#14524002	C₂²	1	1	yes	characteristic
10	#16863445	—	2	4	no	—
11	C₂×#14523979	C₂²	1	4	yes	—
12	#16863354	C₄	1	4	yes	complemented
13	#16863351	C₂²	1	4	yes	—
14	M₁₆×Dic₆₇	C₄	1	4	yes	complemented
15	#16863166	C₂²	1	8	yes	—
16	#16863152	C₂²	1	8	yes	—
17	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#117×C₆₇	C₂²	1	2	yes	—
19	#428×C₆₇	—	2	4	no	—
20	#427×C₆₇	C₂²	1	4	yes	—
21	C₂×#11761612	—	2	2	no	—
22	C₂×#11761612	—	2	2	no	—
23	C₂×#11761612	—	2	2	no	—
24	C₂×#11761612	—	2	2	no	—
25	C₂×#11761612	—	2	2	no	—
26	C₂×#11761612	—	2	2	no	—
27	C₂×#11761612	—	2	2	no	—
28	C₂×#11761612	C₂×C₄	1	2	yes	—
29	C₂×#11761612	C₂³	1	4	yes	—
30	C₂×#11761612	C₂³	1	2	yes	—
31	C₂²×C₆₇⋊C₈	—	2	2	no	—
32	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
33	#14524002	—	2	1	no	—
34	#14524002	—	2	1	no	—
35	#14524002	C₂×C₄	1	2	yes	—
36	#14524002	C₂³	1	2	yes	—
37	C₂×C₄⋊Dic₆₇	—	2	1	no	—
38	C₂×C₂²⋊Dic₆₇	—	2	1	no	—
39	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
40	#14523979	—	2	8	no	—
41	#14523953	C₂×C₄	1	8	yes	complemented
42	C₈×Dic₆₇	C₂×C₄	1	8	yes	complemented
43	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
44	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
45	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
46	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
47	C₂×M₁₆×C₆₇	Q₈	1	4	yes	—
48	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
49	#117×C₆₇	—	2	2	no	—
50	#117×C₆₇	—	2	2	no	—
51	#117×C₆₇	—	2	2	no	—
52	#117×C₆₇	C₂³	1	4	yes	—
53	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
54	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
55	C₂×C₄²×C₆₇	—	2	2	no	—
56	#107×C₆₇	—	2	4	no	—
57	#106×C₆₇	—	2	4	no	—
58	C₂²×Dic₆₇	—	2	2	no	—
59	#11761612	—	2	4	no	—
60	#11761612	—	2	2	no	—
61	#11761612	—	2	2	no	—
62	#11761612	—	2	4	no	—
63	#11761612	—	2	4	no	—
64	#11761612	—	4	2	no	—
65	#11761612	—	4	2	no	—
66	C₂×C₆₇⋊C₈	—	2	4	no	—
67	C₂×C₆₇⋊C₈	—	2	4	no	—
68	C₂×C₆₇⋊C₈	—	4	2	no	—
69	C₂×C₆₇⋊C₈	—	4	2	no	—
70	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
71	C₄⋊Dic₆₇	—	2	2	no	—
72	C₄⋊Dic₆₇	—	2	2	no	—
73	C₂²⋊Dic₆₇	—	2	2	no	—
74	C₂²⋊Dic₆₇	—	2	2	no	—
75	C₄×Dic₆₇	—	2	2	no	—
76	C₄×Dic₆₇	C₂²×C₄	1	4	yes	—
77	C₂⁴×C₆₇	#41	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	—	4	2	no	—
79	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
83	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	#41	1	2	yes	—
85	C₂²×C₄×C₆₇	#41	1	2	yes	—
86	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
89	M₁₆×C₆₇	C₄⋊C₄	1	8	yes	—
90	M₁₆×C₆₇	C₄⋊C₄	1	8	yes	—
91	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
92	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
93	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
94	C₄⋊C₄×C₆₇	—	2	4	no	—
95	C₄⋊C₄×C₆₇	—	4	2	no	—
96	C₂²⋊C₄×C₆₇	—	2	4	no	—
97	C₂²⋊C₄×C₆₇	—	4	2	no	—
98	C₄²×C₆₇	—	2	4	no	—
99	C₄²×C₆₇	—	4	2	no	—
100	C₂³×C₆₇	—	8	1	no	—
101	C₂³×C₆₇	#117	1	1	yes	characteristic
102	C₂³×C₆₇	#117	1	1	yes	characteristic
103	C₂³×C₆₇	#124	1	2	yes	—
104	C₂³×C₆₇	#125	1	2	yes	—
105	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
106	C₂×C₄×C₆₇	—	4	4	no	—
107	C₂×C₄×C₆₇	—	8	4	no	—
108	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₆₇	#117	1	2	yes	—
111	C₂×C₄×C₆₇	#117	1	2	yes	—
112	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
113	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
114	C₂×C₄×C₆₇	#124	1	2	yes	—
115	C₂×C₄×C₆₇	#124	1	2	yes	—
116	C₂×C₄×C₆₇	#124	1	2	yes	—
117	C₂×C₄×C₆₇	#125	1	2	yes	—
118	C₂×C₄×C₆₇	#125	1	2	yes	—
119	C₂×C₄×C₆₇	#125	1	2	yes	—
120	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
126	C₂×Dic₆₇	—	2	4	no	—
127	C₂×Dic₆₇	—	8	2	no	—
128	C₈×C₆₇	C₂×C₄⋊C₄	1	8	yes	—
129	C₆₇⋊C₈	—	2	4	no	—
130	C₆₇⋊C₈	—	4	4	no	—
131	C₂²×C₆₇	—	4	1	no	—
132	C₂²×C₆₇	—	4	2	no	—
133	C₂²×C₆₇	—	4	1	no	—
134	C₂²×C₆₇	—	4	2	no	—
135	C₂²×C₆₇	—	4	1	no	—
136	C₂²×C₆₇	#382	1	2	yes	—
137	C₂²×C₆₇	#382	1	2	yes	—
138	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
139	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
140	C₂²×C₆₇	C₂×#125	1	1	yes	characteristic
141	C₄×C₆₇	—	8	4	no	—
142	C₄×C₆₇	#382	1	2	yes	—
143	C₄×C₆₇	#382	1	2	yes	—
144	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
145	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
146	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
147	C₄×C₆₇	C₂×#125	1	1	yes	characteristic
148	Dic₆₇	—	4	4	no	—
149	#20757	—	67	1	no	maximal, Sylow
150	C₂×C₆₇	—	2	1	no	—
151	C₂×C₆₇	—	2	1	no	—
152	C₂×C₆₇	—	2	1	no	—
153	C₂×C₆₇	—	2	2	no	—
154	C₂×C₆₇	—	2	2	no	—
155	C₂×C₆₇	C₂×#382	1	1	yes	characteristic
156	C₂×#430	—	67	1	no	—
157	#2911	D₆₇	1	1	yes	characteristic
158	#2873	—	67	1	no	—
159	#1833	—	67	8	no	—
160	C₂×#344	—	67	2	no	—
161	C₂×H(3,ℤ/4)	—	67	2	no	—
162	C₆₇	#20757	1	1	yes	characteristic, complemented, Sylow
163	C₂²×M₁₆	—	67	1	no	—
164	C₂²×M₁₆	—	67	2	no	—
165	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
166	C₂×#117	—	67	1	no	—
167	C₂×#117	C₂×D₆₇	1	2	yes	—
168	#430	—	134	4	no	—
169	C₂×#108	—	67	4	no	—
170	#428	—	2	4	no	—
171	#427	C₂×D₆₇	1	4	yes	—
172	#405	—	67	4	no	—
173	#405	—	67	4	no	—
174	C₄×M₁₆	—	67	4	no	—
175	#344	—	67	8	no	—
176	H(3,ℤ/4)	—	67	8	no	—
177	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
178	C₂×M₁₆	—	67	2	no	—
179	C₂×M₁₆	—	67	2	no	—
180	C₂×M₁₆	—	67	4	no	—
181	C₂×M₁₆	—	134	2	no	—
182	C₂×M₁₆	—	134	2	no	—
183	C₂×M₁₆	—	134	2	no	—
184	C₂×M₁₆	—	134	2	no	—
185	C₂×M₁₆	—	134	2	no	—
186	C₂×M₁₆	—	134	2	no	—
187	C₂×M₁₆	—	134	2	no	—
188	C₂×M₁₆	Dic₁₃₄	1	4	yes	—
189	C₂×M₁₆	C₄×D₆₇	1	2	yes	—
190	C₂×M₁₆	C₄×D₆₇	1	2	yes	—
191	C₂×M₁₆	C₂²⋊D₆₇	1	4	yes	—
192	C₂²×C₈	—	67	2	no	—
193	C₂²×C₈	—	134	2	no	—
194	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
195	#117	—	2	2	no	—
196	#117	—	2	2	no	—
197	#117	—	2	2	no	—
198	#117	—	67	2	no	—
199	#117	—	67	2	no	—
200	#117	—	134	1	no	—
201	#117	—	134	1	no	—
202	#117	C₂²×D₆₇	1	4	yes	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	134	1	no	—
205	C₂×C₂²⋊C₄	—	2	2	no	—
206	C₂×C₂²⋊C₄	—	134	1	no	—
207	C₂×C₄²	—	2	2	no	—
208	C₂×C₄²	—	67	2	no	—
209	#108	—	134	8	no	—
210	#107	—	2	4	no	—
211	#106	—	2	4	no	—
212	#97	—	67	8	no	—
213	C₄×C₈	—	67	8	no	—
214	C₂⁴	#11761627	1	1	yes	characteristic
215	C₂²×C₄	—	4	2	no	—
216	C₂²×C₄	—	134	2	no	—
217	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
220	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
221	C₂²×C₄	#11761627	1	1	yes	characteristic
222	C₂²×C₄	#11761627	1	2	yes	—
223	C₂²×C₄	#11761629	1	2	yes	—
224	C₂²×C₄	#11761629	1	1	yes	characteristic
225	C₂²×C₄	#11761629	1	1	yes	characteristic
226	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
227	M₁₆	—	268	2	no	—
228	M₁₆	—	268	2	no	—
229	M₁₆	—	134	4	no	—
230	M₁₆	—	134	4	no	—
231	M₁₆	—	134	4	no	—
232	M₁₆	—	134	2	no	—
233	M₁₆	—	134	2	no	—
234	M₁₆	Dic₆₇⋊C₄	1	8	yes	—
235	M₁₆	Dic₆₇⋊C₄	1	8	yes	—
236	C₂×C₈	—	268	2	no	—
237	C₂×C₈	—	268	2	no	—
238	C₂×C₈	—	67	4	no	—
239	C₂×C₈	—	134	4	no	—
240	C₂×C₈	—	134	4	no	—
241	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
242	C₂×C₈	C₂×C₂²⋊D₆₇	1	4	yes	—
243	C₂×C₈	C₂×Dic₁₃₄	1	4	yes	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	134	2	no	—
247	C₄⋊C₄	—	134	2	no	—
248	C₂²⋊C₄	—	2	4	no	—
249	C₂²⋊C₄	—	4	2	no	—
250	C₂²⋊C₄	—	134	2	no	—
251	C₂²⋊C₄	—	134	2	no	—
252	C₄²	—	2	4	no	—
253	C₄²	—	4	2	no	—
254	C₄²	—	67	4	no	—
255	C₄²	—	134	2	no	—
256	C₂³	—	8	1	no	—
257	C₂³	#14524001	1	1	yes	characteristic
258	C₂³	#14524001	1	1	yes	characteristic
259	C₂³	#14524030	1	2	yes	—
260	C₂³	#14524034	1	2	yes	—
261	C₂³	C₂×#11761627	1	1	yes	characteristic
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	8	4	no	—
264	C₂×C₄	—	536	2	no	—
265	C₂×C₄	—	134	4	no	—
266	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
267	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
268	C₂×C₄	#14524001	1	1	yes	characteristic
269	C₂×C₄	#14524001	1	1	yes	characteristic
270	C₂×C₄	#14524001	1	2	yes	—
271	C₂×C₄	#14524001	1	2	yes	—
272	C₂×C₄	#14524030	1	2	yes	—
273	C₂×C₄	#14524032	1	2	yes	—
274	C₂×C₄	#14524032	1	2	yes	—
275	C₂×C₄	#14524034	1	2	yes	—
276	C₂×C₄	#14524034	1	2	yes	—
277	C₂×C₄	#14524034	1	2	yes	—
278	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
279	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
280	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
281	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
282	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
283	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
284	C₈	—	268	4	no	—
285	C₈	—	134	4	no	—
286	C₈	C₂×Dic₆₇⋊C₄	1	8	yes	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	1	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	2	no	—
291	C₂²	—	4	1	no	—
292	C₂²	#16863261	1	2	yes	—
293	C₂²	#16863261	1	2	yes	—
294	C₂²	C₂×#14524001	1	1	yes	characteristic
295	C₂²	C₂×#14524030	1	1	yes	characteristic
296	C₂²	C₂×#14524034	1	1	yes	characteristic
297	C₄	—	8	4	no	—
298	C₄	—	268	4	no	—
299	C₄	#16863261	1	2	yes	—
300	C₄	#16863261	1	2	yes	—
301	C₄	C₂²×Dic₆₇⋊C₄	1	1	yes	characteristic
302	C₄	C₂×#14524001	1	1	yes	characteristic
303	C₄	C₂×#14524032	1	1	yes	characteristic
304	C₄	C₂×#14524034	1	1	yes	characteristic
305	C₂	—	2	1	no	—
306	C₂	—	2	1	no	—
307	C₂	—	2	2	no	—
308	C₂	—	2	2	no	—
309	C₂	—	2	1	no	—
310	C₂	C₂×#16863261	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#14524001
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#2911	C₂
socle	C₂×C₆₇	C₂×#382
prime radical	C₂⁴×C₆₇	#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#2911	D₆₇
hypocenter	C₆₇	#20757
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #2911

Complements

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

N	H ≅ G/N	type
3. #22560332	C₂	semidirect
12. #16863354	C₄	semidirect
14. M₁₆×Dic₆₇	C₄	semidirect
41. #14523953	C₂×C₄	semidirect
42. C₈×Dic₆₇	C₂×C₄	semidirect
162. C₆₇	#20757	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863445	2
#22568627	2
#22560332	2
C₂×#16863166	2
C₂×#16863152	2
C₆₇×#2911	2
#20757	67