Subgroups — #30859405

Order 17152.

844 conjugacy classes of subgroups.

The subgroup lattice has 314 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	C₂×#16863445	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863444	C₂	1	2	yes	maximal
4	#22560999	C₂	1	8	yes	maximal, complemented
5	C₂×#16863166	C₂	1	2	yes	maximal
6	C₆₇×#2873	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11761612	C₂²	1	2	yes	—
8	C₂²×#11761612	C₂²	1	2	yes	—
9	#16863445	—	2	2	no	—
10	#16863445	—	2	2	no	—
11	#16863445	C₄	1	8	yes	complemented
12	#16863444	—	2	4	no	—
13	#16863444	C₂²	1	8	yes	—
14	C₂×#14523979	C₂²	1	4	yes	—
15	C₂×#14523979	C₂²	1	4	yes	—
16	C₂×#14523978	—	2	4	no	—
17	#16863166	C₂²	1	8	yes	—
18	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
21	#405×C₆₇	—	2	2	no	—
22	#405×C₆₇	C₂²	1	4	yes	—
23	C₄×M₁₆×C₆₇	—	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	—	2	2	no	—
29	C₂×#11761612	—	2	2	no	—
30	C₂×#11761612	—	2	2	no	—
31	C₂×#11761612	—	2	2	no	—
32	C₂×#11761612	C₂×C₄	1	4	yes	—
33	C₂×#11761612	C₂×C₄	1	4	yes	—
34	C₂×#11761612	C₂³	1	4	yes	—
35	C₂×#11761612	C₂³	1	4	yes	—
36	C₂²×C₆₇⋊C₈	—	2	2	no	—
37	C₂²×C₆₇⋊C₈	—	2	2	no	—
38	#14523979	—	2	8	no	—
39	#14523979	C₂×C₄	1	16	yes	complemented
40	#14523978	—	2	16	no	—
41	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
42	C₂×M₁₆×C₆₇	—	2	1	no	—
43	C₂×M₁₆×C₆₇	—	2	2	no	—
44	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
45	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
46	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
47	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
48	C₂×M₁₆×C₆₇	Q₈	1	4	yes	—
49	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
50	C₂²×C₈×C₆₇	—	2	1	no	—
51	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
52	#117×C₆₇	—	2	1	no	—
53	#117×C₆₇	—	2	1	no	—
54	#117×C₆₇	Q₈	1	2	yes	—
55	#117×C₆₇	C₂³	1	2	yes	—
56	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
57	C₂×C₂²⋊C₄×C₆₇	Q₈	1	2	yes	—
58	C₂×C₄²×C₆₇	—	2	1	no	—
59	#97×C₆₇	—	2	4	no	—
60	C₄×C₈×C₆₇	—	2	4	no	—
61	#11761612	—	2	4	no	—
62	#11761612	—	2	4	no	—
63	#11761612	—	2	4	no	—
64	#11761612	—	2	4	no	—
65	#11761612	—	2	4	no	—
66	#11761612	—	4	2	no	—
67	#11761612	—	4	2	no	—
68	#11761612	C₂²×C₄	1	8	yes	—
69	C₂×C₆₇⋊C₈	—	2	4	no	—
70	C₂×C₆₇⋊C₈	—	2	4	no	—
71	C₂×C₆₇⋊C₈	—	2	4	no	—
72	C₂×C₆₇⋊C₈	—	2	4	no	—
73	C₂×C₆₇⋊C₈	—	4	2	no	—
74	C₂×C₆₇⋊C₈	—	4	2	no	—
75	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	—	2	2	no	—
77	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	#41	1	2	yes	—
86	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
87	M₁₆×C₆₇	—	2	4	no	—
88	M₁₆×C₆₇	—	2	2	no	—
89	M₁₆×C₆₇	C₄⋊C₄	1	8	yes	—
90	M₁₆×C₆₇	C₄⋊C₄	1	8	yes	—
91	M₁₆×C₆₇	C₂×D₄	1	4	yes	—
92	C₂×C₈×C₆₇	—	2	4	no	—
93	C₂×C₈×C₆₇	—	2	2	no	—
94	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
95	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
96	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
97	C₄⋊C₄×C₆₇	—	2	2	no	—
98	C₄⋊C₄×C₆₇	—	2	2	no	—
99	C₂²⋊C₄×C₆₇	—	2	2	no	—
100	C₂²⋊C₄×C₆₇	C₂×Q₈	1	4	yes	—
101	C₄²×C₆₇	—	2	2	no	—
102	C₄²×C₆₇	—	2	2	no	—
103	C₂³×C₆₇	—	8	1	no	—
104	C₂³×C₆₇	C₄×Q₈	1	2	yes	—
105	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂³×C₆₇	C₄⋊Q₈	1	2	yes	—
108	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₆₇	—	2	4	no	—
110	C₂×C₄×C₆₇	—	8	2	no	—
111	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
116	C₂×C₄×C₆₇	#121	1	2	yes	—
117	C₂×C₄×C₆₇	#121	1	2	yes	—
118	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₆₇	#125	1	2	yes	—
121	C₂×C₄×C₆₇	#125	1	2	yes	—
122	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
123	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
129	C₈×C₆₇	—	2	4	no	—
130	C₈×C₆₇	C₂×C₄⋊C₄	1	8	yes	—
131	C₆₇⋊C₈	—	2	8	no	—
132	C₆₇⋊C₈	—	4	4	no	—
133	C₂²×C₆₇	—	4	1	no	—
134	C₂²×C₆₇	—	4	2	no	—
135	C₂²×C₆₇	—	4	1	no	—
136	C₂²×C₆₇	—	4	2	no	—
137	C₂²×C₆₇	—	4	1	no	—
138	C₂²×C₆₇	#389	1	2	yes	—
139	C₂²×C₆₇	#389	1	2	yes	—
140	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
141	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
142	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
143	C₄×C₆₇	—	4	4	no	—
144	C₄×C₆₇	#389	1	2	yes	—
145	C₄×C₆₇	#389	1	2	yes	—
146	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
147	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
148	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
149	C₄×C₆₇	C₂×#125	1	1	yes	characteristic
150	#21184	—	67	1	no	maximal, Sylow
151	C₂×C₆₇	—	2	1	no	—
152	C₂×C₆₇	—	2	1	no	—
153	C₂×C₆₇	—	2	1	no	—
154	C₂×C₆₇	—	2	2	no	—
155	C₂×C₆₇	—	2	2	no	—
156	C₂×C₆₇	C₂×#389	1	1	yes	characteristic
157	C₂×#430	—	67	1	no	—
158	C₂×#430	—	67	1	no	—
159	C₂×#430	—	67	2	no	—
160	#2873	D₆₇	1	1	yes	characteristic
161	#1953	—	67	8	no	—
162	C₂×#344	—	67	2	no	—
163	C₆₇	#21184	1	1	yes	characteristic, complemented, Sylow
164	C₂²×M₁₆	—	67	2	no	—
165	C₂²×M₁₆	—	67	2	no	—
166	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
167	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
168	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
169	#430	—	67	8	no	—
170	#430	—	67	8	no	—
171	#430	—	134	4	no	—
172	#430	—	134	2	no	—
173	#430	—	134	2	no	—
174	C₂×#108	—	67	4	no	—
175	C₂×#108	—	67	4	no	—
176	C₂×#108	—	134	4	no	—
177	#405	—	2	2	no	—
178	#405	C₂×D₆₇	1	4	yes	—
179	C₄×M₁₆	—	2	2	no	—
180	#344	—	67	8	no	—
181	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
182	C₂×M₁₆	—	2	1	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	67	4	no	—
185	C₂×M₁₆	—	67	4	no	—
186	C₂×M₁₆	—	67	4	no	—
187	C₂×M₁₆	—	67	4	no	—
188	C₂×M₁₆	—	134	2	no	—
189	C₂×M₁₆	—	134	2	no	—
190	C₂×M₁₆	—	134	2	no	—
191	C₂×M₁₆	—	134	2	no	—
192	C₂×M₁₆	—	134	2	no	—
193	C₂×M₁₆	—	134	2	no	—
194	C₂×M₁₆	—	134	2	no	—
195	C₂×M₁₆	—	134	2	no	—
196	C₂×M₁₆	Dic₁₃₄	1	4	yes	—
197	C₂×M₁₆	C₄×D₆₇	1	4	yes	—
198	C₂×M₁₆	D₂₆₈	1	4	yes	—
199	C₂×M₁₆	C₂²⋊D₆₇	1	2	yes	—
200	C₂×M₁₆	C₂²⋊D₆₇	1	2	yes	—
201	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
202	C₂²×C₈	—	2	1	no	—
203	C₂²×C₈	—	134	2	no	—
204	C₂²×C₈	—	134	2	no	—
205	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
206	#117	—	2	1	no	—
207	#117	—	2	1	no	—
208	#117	Dic₁₃₄	1	2	yes	—
209	#117	C₂²×D₆₇	1	2	yes	—
210	C₂×C₄⋊C₄	—	2	1	no	—
211	C₂×C₂²⋊C₄	Dic₁₃₄	1	2	yes	—
212	C₂×C₄²	—	2	1	no	—
213	#108	—	67	16	no	—
214	#108	—	134	16	no	—
215	#108	—	134	8	no	—
216	#97	—	2	4	no	—
217	C₄×C₈	—	2	4	no	—
218	C₂⁴	C₂×Dic₁₃₄	1	1	yes	characteristic
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
221	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
224	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
225	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
226	C₂²×C₄	#11761628	1	1	yes	characteristic
227	C₂²×C₄	#11761628	1	1	yes	characteristic
228	C₂²×C₄	#11761628	1	2	yes	—
229	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
230	M₁₆	—	2	4	no	—
231	M₁₆	—	2	2	no	—
232	M₁₆	—	268	2	no	—
233	M₁₆	—	268	2	no	—
234	M₁₆	—	67	8	no	—
235	M₁₆	—	134	4	no	—
236	M₁₆	—	134	4	no	—
237	M₁₆	—	134	4	no	—
238	M₁₆	—	134	4	no	—
239	M₁₆	—	134	4	no	—
240	M₁₆	Dic₆₇⋊C₄	1	8	yes	—
241	M₁₆	Dic₆₇⋊C₄	1	8	yes	—
242	M₁₆	C₂×D₂₆₈	1	4	yes	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	268	2	no	—
246	C₂×C₈	—	268	2	no	—
247	C₂×C₈	—	134	4	no	—
248	C₂×C₈	—	134	4	no	—
249	C₂×C₈	—	134	4	no	—
250	C₂×C₈	—	134	4	no	—
251	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
252	C₂×C₈	C₂×C₂²⋊D₆₇	1	4	yes	—
253	C₂×C₈	C₂×Dic₁₃₄	1	4	yes	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₂²⋊C₄	—	2	2	no	—
257	C₂²⋊C₄	C₂×Dic₁₃₄	1	4	yes	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₂³	—	8	1	no	—
261	C₂³	C₄×Dic₁₃₄	1	2	yes	—
262	C₂³	#14524022	1	1	yes	characteristic
263	C₂³	#14524022	1	1	yes	characteristic
264	C₂³	#14524043	1	2	yes	—
265	C₂³	C₂²×Dic₁₃₄	1	1	yes	characteristic
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	8	2	no	—
268	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
269	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
270	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
271	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
272	C₂×C₄	C₄×Dic₁₃₄	1	2	yes	—
273	C₂×C₄	#14524017	1	2	yes	—
274	C₂×C₄	#14524017	1	2	yes	—
275	C₂×C₄	#14524022	1	1	yes	characteristic
276	C₂×C₄	#14524022	1	1	yes	characteristic
277	C₂×C₄	#14524035	1	2	yes	—
278	C₂×C₄	#14524035	1	2	yes	—
279	C₂×C₄	#14524043	1	2	yes	—
280	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
281	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
282	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
283	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
284	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
285	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
286	C₈	—	2	4	no	—
287	C₈	—	268	4	no	—
288	C₈	—	134	8	no	—
289	C₈	C₂×Dic₆₇⋊C₄	1	8	yes	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	2	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	4	2	no	—
294	C₂²	—	4	1	no	—
295	C₂²	#16863294	1	2	yes	—
296	C₂²	#16863294	1	2	yes	—
297	C₂²	C₂×C₄×Dic₁₃₄	1	1	yes	characteristic
298	C₂²	C₂×#14524022	1	1	yes	characteristic
299	C₂²	C₂×#14524043	1	1	yes	characteristic
300	C₄	—	4	4	no	—
301	C₄	#16863294	1	2	yes	—
302	C₄	#16863294	1	2	yes	—
303	C₄	C₂²×Dic₆₇⋊C₄	1	1	yes	characteristic
304	C₄	C₂×C₄×D₂₆₈	1	1	yes	characteristic
305	C₄	C₂×#14524017	1	1	yes	characteristic
306	C₄	C₂×#14524035	1	1	yes	characteristic
307	C₂	—	2	1	no	—
308	C₂	—	2	2	no	—
309	C₂	—	2	2	no	—
310	C₂	—	2	1	no	—
311	C₂	—	2	1	no	—
312	C₂	C₂×#16863294	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#14524017
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#2873	C₂
socle	C₂×C₆₇	C₂×#389
prime radical	C₂⁴×C₆₇	C₂×Q₈
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#2873	D₆₇
hypocenter	C₆₇	#21184
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #2873

Complements

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

N	H ≅ G/N	type
4. #22560999	C₂	semidirect
11. #16863445	C₄	semidirect
39. #14523979	C₂×C₄	semidirect
163. C₆₇	#21184	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
C₂×#16863444	2
#22560999	2
C₂×#16863166	2
C₆₇×#2873	2
#21184	67