Subgroups — #30767318

Order 17152.

314 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#22558968	C₂	1	1	yes	maximal, characteristic
3	#22558267	C₂	1	2	yes	maximal, complemented
4	#22558258	C₂	1	2	yes	maximal, complemented
5	C₆₇×#3170	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₁₆	—	2	1	no	—
7	#16863420	—	2	1	no	—
8	C₄⋊(C₆₇⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
10	#16863149	—	2	1	no	—
11	#16863149	C₂²	1	2	yes	—
12	#533×C₆₇	—	2	1	no	—
13	#500×C₆₇	C₂²	1	1	yes	characteristic
14	#494×C₆₇	C₂²	1	1	yes	characteristic
15	#487×C₆₇	—	2	1	no	—
16	#467×C₆₇	—	2	1	no	—
17	#467×C₆₇	C₄	1	2	yes	—
18	C₄×Q₁₆×C₆₇	—	2	1	no	—
19	C₄×Q₁₆×C₆₇	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#11761612	—	4	1	no	—
22	C₂×C₆₇⋊C₁₆	—	2	2	no	—
23	C₂×C₆₇⋊C₁₆	—	4	2	no	—
24	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	—	4	1	no	—
26	C₂×Q₁₆×C₆₇	—	2	1	no	—
27	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
30	C₄⋊Q₈×C₆₇	—	2	1	no	—
31	C₄⋊Q₈×C₆₇	—	2	1	no	—
32	#125×C₆₇	—	4	1	no	—
33	C₂²⋊Q₈×C₆₇	—	2	2	no	—
34	C₂²⋊Q₈×C₆₇	—	4	1	no	—
35	C₄×Q₈×C₆₇	—	2	2	no	—
36	C₄×Q₈×C₆₇	—	4	1	no	—
37	#117×C₆₇	—	4	1	no	—
38	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
39	#107×C₆₇	—	2	1	no	—
40	#107×C₆₇	C₂×C₄	1	2	yes	—
41	#103×C₆₇	—	2	2	no	—
42	#103×C₆₇	—	2	2	no	—
43	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
45	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
46	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
47	#11761612	—	8	2	no	—
48	C₂×C₆₇⋊C₈	—	4	2	no	—
49	C₆₇⋊C₁₆	—	4	2	no	—
50	C₂×Q₈×C₆₇	—	2	2	no	—
51	C₂×Q₈×C₆₇	—	4	1	no	—
52	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
55	Q₁₆×C₆₇	—	4	2	no	—
56	Q₁₆×C₆₇	—	4	1	no	—
57	Q₁₆×C₆₇	—	4	1	no	—
58	C₂×C₈×C₆₇	—	2	1	no	—
59	C₂×C₈×C₆₇	—	2	1	no	—
60	C₂×C₈×C₆₇	—	4	1	no	—
61	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₄⋊C₄×C₆₇	—	2	2	no	—
66	C₄⋊C₄×C₆₇	—	4	2	no	—
67	C₄⋊C₄×C₆₇	—	4	2	no	—
68	C₄⋊C₄×C₆₇	—	4	1	no	—
69	C₄⋊C₄×C₆₇	—	4	1	no	—
70	C₄⋊C₄×C₆₇	—	4	1	no	—
71	C₂²⋊C₄×C₆₇	—	4	2	no	—
72	C₄²×C₆₇	—	4	2	no	—
73	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₄²×C₆₇	D₈	1	1	yes	characteristic
75	C₄²×C₆₇	SD₁₆	1	1	yes	characteristic
76	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₂³×C₆₇	#102	1	1	yes	characteristic
78	Q₈×C₆₇	—	4	2	no	—
79	Q₈×C₆₇	—	4	2	no	—
80	Q₈×C₆₇	—	8	2	no	—
81	C₂×C₄×C₆₇	—	2	1	no	—
82	C₂×C₄×C₆₇	—	2	1	no	—
83	C₂×C₄×C₆₇	—	4	1	no	—
84	C₂×C₄×C₆₇	—	4	2	no	—
85	C₂×C₄×C₆₇	—	4	2	no	—
86	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
87	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
88	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
89	C₂×C₄×C₆₇	SD₃₂	1	2	yes	—
90	C₂×C₄×C₆₇	Q₃₂	1	2	yes	—
91	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
94	C₈×C₆₇	—	2	1	no	—
95	C₈×C₆₇	—	2	1	no	—
96	C₈×C₆₇	—	2	1	no	—
97	C₈×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
98	C₈×C₆₇	#137	1	1	yes	characteristic
99	C₆₇⋊C₈	—	8	2	no	—
100	C₂²×C₆₇	—	2	1	no	—
101	C₂²×C₆₇	—	2	1	no	—
102	C₂²×C₆₇	#358	1	2	yes	—
103	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
104	C₄×C₆₇	—	2	1	no	—
105	C₄×C₆₇	—	2	1	no	—
106	C₄×C₆₇	—	2	1	no	—
107	C₄×C₆₇	—	8	2	no	—
108	C₄×C₆₇	—	8	2	no	—
109	C₄×C₆₇	#358	1	2	yes	—
110	C₄×C₆₇	#413	1	1	yes	characteristic
111	C₄×C₆₇	#417	1	1	yes	characteristic
112	C₄×C₆₇	C₂×SD₃₂	1	1	yes	characteristic
113	C₄×C₆₇	C₂×Q₃₂	1	1	yes	characteristic
114	#11983	—	67	1	no	maximal, Sylow
115	C₂×C₆₇	—	2	2	no	—
116	C₂×C₆₇	#1547	1	1	yes	characteristic
117	C₂×C₆₇	C₂×#358	1	1	yes	characteristic
118	C₂×C₆₇	#2148	1	1	yes	characteristic
119	#3170	D₆₇	1	1	yes	characteristic
120	C₂×C₄⋊C₁₆	—	67	1	no	—
121	#1567	—	67	1	no	—
122	#1365	—	67	2	no	—
123	#1362	—	67	2	no	—
124	C₆₇	#11983	1	1	yes	characteristic, complemented, Sylow
125	#533	—	2	1	no	—
126	C₂²×C₁₆	—	134	1	no	—
127	#500	C₂×D₆₇	1	1	yes	characteristic
128	#494	C₂×D₆₇	1	1	yes	characteristic
129	#487	—	2	1	no	—
130	#467	—	2	1	no	—
131	#467	Dic₆₇	1	2	yes	—
132	C₄×Q₁₆	—	2	1	no	—
133	C₄×Q₁₆	Dic₆₇	1	2	yes	—
134	#423	—	134	1	no	—
135	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
136	C₄⋊C₁₆	—	67	2	no	—
137	C₄⋊C₁₆	—	134	1	no	—
138	#335	—	67	2	no	—
139	#335	—	134	1	no	—
140	C₂×Q₁₆	—	2	1	no	—
141	C₂×Q₁₆	C₂×Dic₆₇	1	2	yes	—
142	C₂×M₁₆	—	268	1	no	—
143	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
144	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
145	C₄⋊Q₈	—	2	1	no	—
146	C₄⋊Q₈	—	2	1	no	—
147	#125	—	4	1	no	—
148	C₂²⋊Q₈	—	2	2	no	—
149	C₂²⋊Q₈	—	4	1	no	—
150	C₄×Q₈	—	2	2	no	—
151	C₄×Q₈	—	4	1	no	—
152	#117	—	4	1	no	—
153	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
154	C₂×C₁₆	—	268	2	no	—
155	C₂×C₁₆	—	134	2	no	—
156	#107	—	2	1	no	—
157	#107	C₂×Dic₆₇	1	2	yes	—
158	C₄⋊C₈	—	268	1	no	—
159	C₄⋊C₈	—	134	2	no	—
160	#103	—	2	2	no	—
161	#103	—	2	2	no	—
162	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
163	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
164	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
165	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
166	C₂×Q₈	—	2	2	no	—
167	C₂×Q₈	—	4	1	no	—
168	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
169	C₂²×C₄	#11761613	1	1	yes	characteristic
170	C₂²×C₄	#11761616	1	1	yes	characteristic
171	Q₁₆	—	4	2	no	—
172	Q₁₆	—	4	1	no	—
173	Q₁₆	—	4	1	no	—
174	M₁₆	—	536	2	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	4	1	no	—
178	C₂×C₈	—	268	2	no	—
179	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
180	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
181	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
182	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	4	2	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄⋊C₄	—	4	1	no	—
187	C₄⋊C₄	—	4	1	no	—
188	C₄⋊C₄	—	4	1	no	—
189	C₂²⋊C₄	—	4	2	no	—
190	C₄²	—	4	2	no	—
191	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
192	C₄²	#11761613	1	1	yes	characteristic
193	C₄²	#11761616	1	1	yes	characteristic
194	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
195	C₁₆	—	268	2	no	—
196	C₂³	#14523964	1	1	yes	characteristic
197	Q₈	—	4	2	no	—
198	Q₈	—	4	2	no	—
199	Q₈	—	8	2	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	#14523964	1	1	yes	characteristic
206	C₂×C₄	#14523964	1	1	yes	characteristic
207	C₂×C₄	#14523964	1	1	yes	characteristic
208	C₂×C₄	#14523986	1	2	yes	—
209	C₂×C₄	#14523989	1	2	yes	—
210	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
211	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
212	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
213	C₈	—	2	1	no	—
214	C₈	—	2	1	no	—
215	C₈	—	2	1	no	—
216	C₈	—	536	2	no	—
217	C₈	#14524068	1	1	yes	characteristic
218	C₈	#14524073	1	1	yes	characteristic
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	#16863200	1	2	yes	—
222	C₂²	C₂×#14523964	1	1	yes	characteristic
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	8	2	no	—
227	C₄	—	8	2	no	—
228	C₄	#16863200	1	2	yes	—
229	C₄	#16863380	1	1	yes	characteristic
230	C₄	#16863397	1	1	yes	characteristic
231	C₄	C₂×#14523986	1	1	yes	characteristic
232	C₄	C₂×#14523989	1	1	yes	characteristic
233	C₂	—	2	2	no	—
234	C₂	#22558886	1	1	yes	characteristic
235	C₂	C₂×#16863200	1	1	yes	characteristic
236	C₂	#22561958	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14523964
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#3170	C₂
socle	C₂²×C₆₇	C₂×#102
prime radical	C₂³×C₆₇	#102
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	#3170	D₆₇
hypocenter	C₆₇	#11983
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3170

Complements

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

N	H ≅ G/N	type
3. #22558267	C₂	semidirect
4. #22558258	C₂	semidirect
124. C₆₇	#11983	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₆₇⋊C₁₆)	2
#22558968	2
#22558267	2
#22558258	2
C₆₇×#3170	2
#11983	67