Subgroups — #30885802

Order 17152.

664 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	C₂×#16863445	C₂	1	1	yes	maximal, characteristic
2	#22561784	C₂	1	8	yes	maximal, complemented
3	C₂×#16863168	C₂	1	2	yes	maximal
4	#22559919	C₂	1	2	yes	maximal
5	C₂×#16863153	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#2905	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11761612	C₂²	1	2	yes	—
8	C₂×#14524002	C₂²	1	2	yes	—
9	#16863445	—	2	2	no	—
10	#16863445	C₂²	1	4	yes	—
11	C₂×#14523979	—	2	2	no	—
12	#16863168	C₂²	1	8	yes	—
13	#16863156	C₂²	1	8	yes	—
14	#16863153	C₂²	1	4	yes	—
15	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
18	#424×C₆₇	—	2	2	no	—
19	#424×C₆₇	—	2	2	no	—
20	#423×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	C₂³	1	4	yes	—
27	C₂²×C₆₇⋊C₈	—	2	2	no	—
28	#14524002	—	2	2	no	—
29	#14524002	—	2	2	no	—
30	#14524002	—	2	2	no	—
31	#14524002	C₂³	1	4	yes	—
32	C₂×C₄⋊Dic₆₇	—	2	2	no	—
33	C₂×C₂²⋊Dic₆₇	—	2	2	no	—
34	C₂×C₄×Dic₆₇	—	2	2	no	—
35	#14523979	—	2	8	no	—
36	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₆₇	—	2	2	no	—
38	C₂×M₁₆×C₆₇	—	2	2	no	—
39	C₂×M₁₆×C₆₇	—	2	1	no	—
40	C₂×M₁₆×C₆₇	—	2	2	no	—
41	C₂×M₁₆×C₆₇	—	2	2	no	—
42	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
43	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
44	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
45	C₂²×C₈×C₆₇	—	2	1	no	—
46	C₂²×C₈×C₆₇	D₄	1	2	yes	—
47	#117×C₆₇	—	2	1	no	—
48	#117×C₆₇	—	2	1	no	—
49	#117×C₆₇	Q₈	1	2	yes	—
50	#117×C₆₇	Q₈	1	2	yes	—
51	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
52	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
53	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
54	C₄⋊C₈×C₆₇	—	2	8	no	—
55	C₂²×Dic₆₇	—	4	2	no	—
56	#11761612	—	2	4	no	—
57	#11761612	—	2	4	no	—
58	#11761612	—	4	2	no	—
59	#11761612	—	4	2	no	—
60	C₂×C₆₇⋊C₈	—	2	4	no	—
61	C₂×C₆₇⋊C₈	—	4	2	no	—
62	C₂×C₆₇⋊C₈	—	4	2	no	—
63	C₄⋊Dic₆₇	—	2	4	no	—
64	C₄⋊Dic₆₇	—	4	2	no	—
65	C₂²⋊Dic₆₇	—	2	4	no	—
66	C₂²⋊Dic₆₇	—	4	2	no	—
67	C₄×Dic₆₇	—	2	4	no	—
68	C₄×Dic₆₇	—	4	2	no	—
69	C₂⁴×C₆₇	#41	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	—	2	2	no	—
71	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	#41	1	2	yes	—
76	C₂²×C₄×C₆₇	#41	1	2	yes	—
77	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	#41	1	2	yes	—
79	C₂²×C₄×C₆₇	#41	1	2	yes	—
80	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
81	M₁₆×C₆₇	—	2	4	no	—
82	M₁₆×C₆₇	—	2	4	no	—
83	M₁₆×C₆₇	—	2	4	no	—
84	M₁₆×C₆₇	—	4	1	no	—
85	M₁₆×C₆₇	—	4	1	no	—
86	C₂×C₈×C₆₇	—	2	2	no	—
87	C₂×C₈×C₆₇	—	2	2	no	—
88	C₂×C₈×C₆₇	—	2	2	no	—
89	C₂×C₈×C₆₇	—	4	2	no	—
90	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
91	C₄⋊C₄×C₆₇	—	2	2	no	—
92	C₄⋊C₄×C₆₇	—	2	2	no	—
93	C₂²⋊C₄×C₆₇	—	2	2	no	—
94	C₂²⋊C₄×C₆₇	—	2	2	no	—
95	C₄²×C₆₇	—	2	2	no	—
96	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
97	C₂³×C₆₇	—	8	1	no	—
98	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂³×C₆₇	#126	1	2	yes	—
101	C₂³×C₆₇	#126	1	2	yes	—
102	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
103	C₂×C₄×C₆₇	—	2	4	no	—
104	C₂×C₄×C₆₇	—	8	2	no	—
105	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
108	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
109	C₂×C₄×C₆₇	#123	1	2	yes	—
110	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
111	C₂×C₄×C₆₇	#124	1	2	yes	—
112	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
113	C₂×C₄×C₆₇	#125	1	2	yes	—
114	C₂×C₄×C₆₇	#125	1	2	yes	—
115	C₂×C₄×C₆₇	#125	1	2	yes	—
116	C₂×C₄×C₆₇	#125	1	2	yes	—
117	C₂×C₄×C₆₇	#126	1	2	yes	—
118	C₂×C₄×C₆₇	#126	1	2	yes	—
119	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
122	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
123	C₂×Dic₆₇	—	4	4	no	—
124	C₂×Dic₆₇	—	8	4	no	—
125	C₈×C₆₇	—	2	4	no	—
126	C₈×C₆₇	—	4	2	no	—
127	C₆₇⋊C₈	—	4	4	no	—
128	C₂²×C₆₇	—	4	2	no	—
129	C₂²×C₆₇	—	4	1	no	—
130	C₂²×C₆₇	—	4	1	no	—
131	C₂²×C₆₇	—	4	1	no	—
132	C₂²×C₆₇	—	4	2	no	—
133	C₂²×C₆₇	#400	1	1	yes	characteristic
134	C₂²×C₆₇	#400	1	1	yes	characteristic
135	C₂²×C₆₇	#400	1	2	yes	—
136	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
137	C₂²×C₆₇	C₂×#126	1	2	yes	—
138	C₄×C₆₇	—	4	4	no	—
139	C₄×C₆₇	#400	1	1	yes	characteristic
140	C₄×C₆₇	#400	1	1	yes	characteristic
141	C₄×C₆₇	#400	1	2	yes	—
142	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
143	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
144	C₄×C₆₇	C₂×#125	1	2	yes	—
145	Dic₆₇	—	8	4	no	—
146	#23751	—	67	1	no	maximal, Sylow
147	C₂×C₆₇	—	2	2	no	—
148	C₂×C₆₇	—	2	1	no	—
149	C₂×C₆₇	—	2	1	no	—
150	C₂×C₆₇	—	2	2	no	—
151	C₂×C₆₇	—	2	1	no	—
152	C₂×C₆₇	C₂×#400	1	1	yes	characteristic
153	C₂×#430	—	67	1	no	—
154	#2905	D₆₇	1	1	yes	characteristic
155	#2102	—	67	8	no	—
156	C₂×#344	—	67	2	no	—
157	#1733	—	67	2	no	—
158	C₂×H(3,ℤ/4)	—	67	1	no	—
159	C₆₇	#23751	1	1	yes	characteristic, complemented, Sylow
160	C₂²×M₁₆	—	67	2	no	—
161	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
162	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
163	C₂×#117	—	67	2	no	—
164	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
165	#430	—	67	4	no	—
166	#430	—	134	2	no	—
167	C₂×#108	—	134	2	no	—
168	#424	—	2	2	no	—
169	#424	—	2	2	no	—
170	#423	C₂×D₆₇	1	4	yes	—
171	#344	—	67	8	no	—
172	#339	—	67	8	no	—
173	H(3,ℤ/4)	—	67	4	no	—
174	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	1	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	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	C₂²⋊D₆₇	1	2	yes	—
187	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
188	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	—	134	2	no	—
191	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
192	#117	—	2	1	no	—
193	#117	—	2	1	no	—
194	#117	—	67	4	no	—
195	#117	—	134	2	no	—
196	#117	—	134	2	no	—
197	#117	—	134	2	no	—
198	#117	Dic₁₃₄	1	2	yes	—
199	#117	Dic₁₃₄	1	2	yes	—
200	C₂×C₄⋊C₄	—	2	1	no	—
201	C₂×C₄⋊C₄	—	134	2	no	—
202	C₂×C₂²⋊C₄	—	2	1	no	—
203	C₂×C₂²⋊C₄	—	134	2	no	—
204	C₂×C₄²	—	134	2	no	—
205	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
206	#108	—	134	8	no	—
207	C₄⋊C₈	—	2	8	no	—
208	C₂⁴	#11761628	1	1	yes	characteristic
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	268	2	no	—
211	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
213	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
214	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
215	C₂²×C₄	#11761627	1	2	yes	—
216	C₂²×C₄	#11761627	1	2	yes	—
217	C₂²×C₄	#11761628	1	1	yes	characteristic
218	C₂²×C₄	#11761629	1	2	yes	—
219	C₂²×C₄	#11761629	1	2	yes	—
220	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
221	M₁₆	—	2	4	no	—
222	M₁₆	—	2	4	no	—
223	M₁₆	—	2	4	no	—
224	M₁₆	—	4	1	no	—
225	M₁₆	—	4	1	no	—
226	M₁₆	—	268	2	no	—
227	M₁₆	—	268	2	no	—
228	M₁₆	—	134	4	no	—
229	M₁₆	—	134	4	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	268	2	no	—
235	C₂×C₈	—	268	2	no	—
236	C₂×C₈	—	134	4	no	—
237	C₂×C₈	C₂×C₂²⋊D₆₇	1	4	yes	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	268	2	no	—
241	C₄⋊C₄	—	134	4	no	—
242	C₂²⋊C₄	—	2	2	no	—
243	C₂²⋊C₄	—	2	2	no	—
244	C₂²⋊C₄	—	268	2	no	—
245	C₂²⋊C₄	—	134	4	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	268	2	no	—
248	C₄²	—	134	4	no	—
249	C₄²	C₂×Dic₁₃₄	1	4	yes	—
250	C₂³	—	8	1	no	—
251	C₂³	#14524022	1	1	yes	characteristic
252	C₂³	#14524022	1	1	yes	characteristic
253	C₂³	#14524038	1	2	yes	—
254	C₂³	#14524038	1	2	yes	—
255	C₂³	C₂×#11761628	1	1	yes	characteristic
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	8	2	no	—
258	C₂×C₄	—	268	4	no	—
259	C₂×C₄	—	536	4	no	—
260	C₂×C₄	#14524022	1	1	yes	characteristic
261	C₂×C₄	#14524022	1	1	yes	characteristic
262	C₂×C₄	#14524024	1	1	yes	characteristic
263	C₂×C₄	#14524024	1	1	yes	characteristic
264	C₂×C₄	#14524024	1	2	yes	—
265	C₂×C₄	#14524032	1	1	yes	characteristic
266	C₂×C₄	#14524032	1	2	yes	—
267	C₂×C₄	#14524032	1	1	yes	characteristic
268	C₂×C₄	#14524034	1	2	yes	—
269	C₂×C₄	#14524034	1	2	yes	—
270	C₂×C₄	#14524034	1	2	yes	—
271	C₂×C₄	#14524034	1	2	yes	—
272	C₂×C₄	#14524038	1	2	yes	—
273	C₂×C₄	#14524038	1	2	yes	—
274	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
275	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
276	C₂×C₄	C₂×#11761627	1	2	yes	—
277	C₂×C₄	C₂×#11761629	1	2	yes	—
278	C₈	—	2	4	no	—
279	C₈	—	4	2	no	—
280	C₈	—	268	4	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	2	no	—
286	C₂²	#16863335	1	1	yes	characteristic
287	C₂²	#16863335	1	1	yes	characteristic
288	C₂²	#16863335	1	2	yes	—
289	C₂²	C₂×#14524022	1	1	yes	characteristic
290	C₂²	C₂×#14524038	1	2	yes	—
291	C₄	—	4	4	no	—
292	C₄	—	536	4	no	—
293	C₄	#16863335	1	1	yes	characteristic
294	C₄	#16863335	1	1	yes	characteristic
295	C₄	#16863335	1	2	yes	—
296	C₄	C₂×#14524024	1	1	yes	characteristic
297	C₄	C₂×#14524032	1	1	yes	characteristic
298	C₄	C₂×#14524034	1	2	yes	—
299	C₂	—	2	1	no	—
300	C₂	—	2	2	no	—
301	C₂	—	2	1	no	—
302	C₂	—	2	2	no	—
303	C₂	—	2	1	no	—
304	C₂	C₂×#16863335	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#14524032
commutator, cosocle	C₂²×C₄×C₆₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#2905	C₂
socle	C₂×C₆₇	C₂×#400
prime radical	C₂⁴×C₆₇	#41
hypercenter	#2905	D₆₇
hypocenter	C₆₇	#23751
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #2905

Complements

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

N	H ≅ G/N	type
2. #22561784	C₂	semidirect
159. C₆₇	#23751	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
#22561784	2
C₂×#16863168	2
#22559919	2
C₂×#16863153	2
C₆₇×#2905	2
#23751	67