Subgroups — #31008960

Order 17152.

520 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568635	C₂	1	1	yes	maximal, characteristic
2	#22562022	C₂	1	4	yes	maximal, complemented
3	#22562018	C₂	1	4	yes	maximal, complemented
4	#22561974	C₂	1	2	yes	maximal, complemented
5	#22561830	C₂	1	1	yes	maximal, characteristic
6	C₂×#16863173	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2115	C₂	1	1	yes	maximal, characteristic
8	C₂×#14524050	C₄	1	1	yes	characteristic
9	#41×Dic₆₇	C₄	1	1	yes	characteristic
10	C₂×#14523983	C₂²	1	2	yes	—
11	#16863354	C₄	1	1	yes	characteristic
12	#16863354	C₂²	1	2	yes	—
13	#16863351	C₄	1	2	yes	—
14	#16863351	C₂²	1	1	yes	characteristic
15	M₁₆×Dic₆₇	C₄	1	1	yes	characteristic
16	M₁₆×Dic₆₇	C₂²	1	2	yes	—
17	C₂×#14523953	C₄	1	2	yes	—
18	C₂×#14523953	C₂²	1	1	yes	characteristic
19	#16863216	C₂²	1	8	yes	—
20	#16863178	C₂²	1	4	yes	—
21	#16863173	C₂²	1	4	yes	—
22	#16863173	C₂²	1	4	yes	—
23	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
25	#349×C₆₇	C₂²	1	4	yes	—
26	#14524050	—	2	4	no	—
27	C₂×#11761612	C₂×C₄	1	2	yes	—
28	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
31	Q₈×Dic₆₇	—	2	1	no	—
32	D₄×Dic₆₇	—	2	2	no	—
33	D₄×Dic₆₇	—	2	1	no	—
34	#14524002	C₂×C₄	1	2	yes	—
35	#14524002	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
37	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
38	#14523983	—	2	2	no	—
39	#14523983	C₂³	1	4	yes	—
40	C₂×C₆₇⋊C₁₆	—	2	2	no	—
41	#14523953	C₂×C₄	1	1	yes	characteristic
42	#14523953	C₂×C₄	1	2	yes	—
43	#14523953	C₂×C₄	1	2	yes	—
44	#14523953	C₂×C₄	1	2	yes	—
45	#14523953	C₂×C₄	1	1	yes	characteristic
46	#14523953	C₂³	1	1	yes	characteristic
47	#14523953	C₂³	1	1	yes	characteristic
48	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
49	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
50	C₈×Dic₆₇	C₂³	1	2	yes	—
51	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
52	#131×C₆₇	—	2	2	no	—
53	#131×C₆₇	—	2	2	no	—
54	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
56	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
57	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
58	M₃₂×C₆₇	—	2	2	no	—
59	M₃₂×C₆₇	C₂³	1	4	yes	—
60	C₂×C₁₆×C₆₇	—	2	2	no	—
61	C₂²×Dic₆₇	—	2	2	no	—
62	C₂²×Dic₆₇	—	2	1	no	—
63	#11761612	—	2	4	no	—
64	#11761612	M₁₆	1	4	yes	—
65	C₂×C₆₇⋊C₈	—	2	2	no	—
66	C₂×C₆₇⋊C₈	—	2	4	no	—
67	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
70	C₄⋊Dic₆₇	—	2	2	no	—
71	C₄⋊Dic₆₇	—	2	1	no	—
72	C₂²⋊Dic₆₇	—	2	2	no	—
73	C₂²⋊Dic₆₇	—	2	1	no	—
74	C₄×Dic₆₇	—	2	2	no	—
75	C₄×Dic₆₇	—	2	1	no	—
76	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
78	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
79	C₆₇⋊C₁₆	—	2	4	no	—
80	#41×C₆₇	—	2	2	no	—
81	#41×C₆₇	—	2	2	no	—
82	C₂×Q₈×C₆₇	M₁₆	1	1	yes	characteristic
83	C₂×D₄×C₆₇	M₁₆	1	2	yes	—
84	C₂×D₄×C₆₇	M₁₆	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	M₁₆×C₆₇	—	2	2	no	—
88	M₁₆×C₆₇	—	2	4	no	—
89	C₂×C₈×C₆₇	—	2	2	no	—
90	C₂×C₈×C₆₇	—	2	2	no	—
91	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₆₇	#41	1	2	yes	—
94	C₂×C₈×C₆₇	#41	1	2	yes	—
95	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
96	C₁₆×C₆₇	—	2	4	no	—
97	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
98	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
99	Q₈×C₆₇	—	2	2	no	—
100	D₄×C₆₇	—	2	4	no	—
101	D₄×C₆₇	—	2	2	no	—
102	C₂×C₄×C₆₇	—	2	2	no	—
103	C₂×C₄×C₆₇	—	2	2	no	—
104	C₂×C₄×C₆₇	#117	1	2	yes	—
105	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
106	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
107	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
109	C₂×Dic₆₇	—	2	1	no	—
110	C₂×Dic₆₇	—	2	1	no	—
111	C₂×Dic₆₇	—	2	2	no	—
112	C₂×Dic₆₇	—	8	1	no	—
113	C₂×Dic₆₇	—	8	2	no	—
114	C₈×C₆₇	—	2	2	no	—
115	C₈×C₆₇	#117	1	1	yes	characteristic
116	C₈×C₆₇	#117	1	1	yes	characteristic
117	C₈×C₆₇	C₂×#41	1	2	yes	—
118	C₆₇⋊C₈	—	2	1	no	—
119	C₆₇⋊C₈	—	2	1	no	—
120	C₆₇⋊C₈	C₂×M₁₆	1	4	yes	—
121	C₂²×C₆₇	—	2	2	no	—
122	C₂²×C₆₇	—	4	1	no	—
123	C₂²×C₆₇	—	4	2	no	—
124	C₂²×C₆₇	#432	1	2	yes	—
125	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
126	C₄×C₆₇	—	2	2	no	—
127	C₄×C₆₇	#432	1	2	yes	—
128	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
129	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
130	Dic₆₇	—	4	1	no	—
131	Dic₆₇	—	4	2	no	—
132	Dic₆₇	—	4	1	no	—
133	#33426	—	67	1	no	maximal, Sylow
134	C₂×C₆₇	—	2	1	no	—
135	C₂×C₆₇	—	2	2	no	—
136	C₂×C₆₇	—	4	2	no	—
137	C₂×C₆₇	C₂×#432	1	1	yes	characteristic
138	#2875	—	67	1	no	—
139	#2166	—	67	4	no	—
140	#2165	—	67	4	no	—
141	#2153	—	67	2	no	—
142	#2115	—	67	1	no	—
143	#2115	D₆₇	1	1	yes	characteristic
144	C₂×#347	—	67	2	no	—
145	C₆₇	#33426	1	1	yes	characteristic, complemented, Sylow
146	C₂×#131	—	67	1	no	—
147	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
148	C₄×#41	—	67	1	no	—
149	C₂×M₃₂	—	67	2	no	—
150	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
151	#405	—	67	1	no	—
152	#405	—	67	2	no	—
153	#405	—	67	2	no	—
154	#405	—	67	1	no	—
155	C₄×M₁₆	—	67	1	no	—
156	C₄×M₁₆	—	67	2	no	—
157	C₂×#97	—	67	2	no	—
158	C₂×#97	—	67	1	no	—
159	#364	—	67	8	no	—
160	#349	—	67	4	no	—
161	#349	C₂×D₆₇	1	4	yes	—
162	#347	—	67	4	no	—
163	#347	—	67	4	no	—
164	C₂×#41	C₄×D₆₇	1	1	yes	characteristic
165	#131	—	2	2	no	—
166	#131	—	2	2	no	—
167	#131	—	134	4	no	—
168	C₂×M₁₆	—	67	1	no	—
169	C₂×M₁₆	—	67	2	no	—
170	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
171	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
172	C₂²×C₈	—	67	2	no	—
173	C₂²×C₈	—	67	1	no	—
174	C₂²×C₈	C₄×D₆₇	1	2	yes	—
175	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
176	C₄×Q₈	—	134	1	no	—
177	C₄×D₄	—	134	2	no	—
178	C₄×D₄	—	134	1	no	—
179	#117	—	67	2	no	—
180	#117	—	67	1	no	—
181	C₂×C₄²	—	67	2	no	—
182	C₂×C₄²	—	67	1	no	—
183	M₃₂	—	2	2	no	—
184	M₃₂	—	67	4	no	—
185	M₃₂	—	134	2	no	—
186	M₃₂	C₂²×D₆₇	1	4	yes	—
187	C₂×C₁₆	—	2	2	no	—
188	C₂×C₁₆	—	134	2	no	—
189	#97	—	67	1	no	—
190	#97	—	67	2	no	—
191	#97	—	67	2	no	—
192	#97	—	67	1	no	—
193	#97	—	67	2	no	—
194	#97	—	67	1	no	—
195	#97	—	67	1	no	—
196	C₄×C₈	—	67	2	no	—
197	C₄×C₈	—	67	2	no	—
198	C₄×C₈	—	67	2	no	—
199	#41	—	2	2	no	—
200	#41	—	2	2	no	—
201	C₂×Q₈	#11761611	1	1	yes	characteristic
202	C₂×D₄	#11761611	1	2	yes	—
203	C₂×D₄	#11761611	1	1	yes	characteristic
204	C₂²×C₄	—	134	2	no	—
205	C₂²×C₄	—	134	1	no	—
206	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
207	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
208	M₁₆	—	2	2	no	—
209	M₁₆	—	2	4	no	—
210	M₁₆	—	67	4	no	—
211	M₁₆	—	134	4	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	67	2	no	—
215	C₂×C₈	—	67	1	no	—
216	C₂×C₈	—	67	1	no	—
217	C₂×C₈	—	134	4	no	—
218	C₂×C₈	—	134	2	no	—
219	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
220	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
221	C₂×C₈	#11761627	1	2	yes	—
222	C₂×C₈	#11761627	1	2	yes	—
223	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
224	C₄⋊C₄	—	134	2	no	—
225	C₄⋊C₄	—	134	1	no	—
226	C₂²⋊C₄	—	134	2	no	—
227	C₂²⋊C₄	—	134	1	no	—
228	C₄²	—	67	1	no	—
229	C₄²	—	67	2	no	—
230	C₄²	—	67	1	no	—
231	C₄²	—	134	2	no	—
232	C₄²	—	134	1	no	—
233	C₁₆	—	2	4	no	—
234	C₁₆	—	134	4	no	—
235	C₂³	C₂×#11761611	1	2	yes	—
236	C₂³	C₂×#11761611	1	1	yes	characteristic
237	Q₈	—	2	2	no	—
238	D₄	—	2	4	no	—
239	D₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	536	1	no	—
243	C₂×C₄	—	536	2	no	—
244	C₂×C₄	—	134	1	no	—
245	C₂×C₄	—	134	1	no	—
246	C₂×C₄	—	134	2	no	—
247	C₂×C₄	#14523999	1	2	yes	—
248	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
249	C₂×C₄	C₂×#11761611	1	2	yes	—
250	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
251	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
252	C₈	—	2	2	no	—
253	C₈	—	67	4	no	—
254	C₈	—	134	1	no	—
255	C₈	—	134	1	no	—
256	C₈	#14523999	1	1	yes	characteristic
257	C₈	#14523999	1	1	yes	characteristic
258	C₈	C₂×#11761627	1	2	yes	—
259	C₂²	—	2	2	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	2	no	—
262	C₂²	#16863450	1	2	yes	—
263	C₂²	C₂²×#11761611	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	268	1	no	—
266	C₄	—	268	1	no	—
267	C₄	—	268	2	no	—
268	C₄	#16863450	1	2	yes	—
269	C₄	C₂×#14523999	1	1	yes	characteristic
270	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
271	C₂	—	2	1	no	—
272	C₂	—	2	2	no	—
273	C₂	—	4	2	no	—
274	C₂	C₂×#16863450	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#14523999
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2115	C₂
socle	C₂×C₆₇	C₂×#432
prime radical	C₂×#41×C₆₇	C₂×C₄
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	#2115	D₆₇
hypocenter	C₆₇	#33426
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #2115

Complements

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

N	H ≅ G/N	type
2. #22562022	C₂	semidirect
3. #22562018	C₂	semidirect
4. #22561974	C₂	semidirect
6. C₂×#16863173	C₂	semidirect
145. C₆₇	#33426	semidirect

Maximal subgroups

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

subgroup	index
#22568635	2
#22562022	2
#22562018	2
#22561974	2
#22561830	2
C₂×#16863173	2
C₆₇×#2115	2
#33426	67