Subgroups — C₂×#22558267 #30767113

Order 17152.

350 conjugacy classes of subgroups.

The subgroup lattice has 214 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	C₂×#16863149	C₂	1	1	yes	maximal, characteristic
3	#22558267	C₂	1	4	yes	maximal, direct product factor
4	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₆₇⋊C₁₆	—	2	1	no	—
6	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
7	C₄⋊(C₆₇⋊C₁₆)	—	2	1	no	—
8	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
9	#16863149	—	2	1	no	—
10	#16863149	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
12	#500×C₆₇	—	2	1	no	—
13	#500×C₆₇	—	2	1	no	—
14	#500×C₆₇	C₄	1	2	yes	—
15	#500×C₆₇	C₂²	1	2	yes	—
16	C₂×#107×C₆₇	—	2	1	no	—
17	C₂×#107×C₆₇	C₄	1	2	yes	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×C₆₇⋊C₈	—	4	1	no	—
20	C₂×C₆₇⋊C₁₆	—	2	2	no	—
21	C₂×C₆₇⋊C₁₆	—	4	2	no	—
22	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₆₇⋊C₈)	—	4	1	no	—
24	C₂²×Q₈×C₆₇	—	4	1	no	—
25	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
27	C₄⋊Q₈×C₆₇	—	2	2	no	—
28	C₄⋊Q₈×C₆₇	—	2	2	no	—
29	C₄⋊Q₈×C₆₇	—	4	1	no	—
30	C₄⋊Q₈×C₆₇	—	4	1	no	—
31	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
32	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
33	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
34	#107×C₆₇	—	2	2	no	—
35	#107×C₆₇	—	2	2	no	—
36	#107×C₆₇	—	2	2	no	—
37	#107×C₆₇	C₂×C₄	1	4	yes	—
38	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
40	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
41	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
42	C₂×C₆₇⋊C₈	—	4	2	no	—
43	C₂×C₆₇⋊C₈	—	8	2	no	—
44	C₆₇⋊C₁₆	—	4	2	no	—
45	C₂×Q₈×C₆₇	—	4	2	no	—
46	C₂×Q₈×C₆₇	—	4	2	no	—
47	C₂×Q₈×C₆₇	—	8	4	no	—
48	C₂²×C₄×C₆₇	—	4	2	no	—
49	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
52	C₂×C₈×C₆₇	—	2	2	no	—
53	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
56	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
57	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
59	C₄⋊C₄×C₆₇	—	2	4	no	—
60	C₄⋊C₄×C₆₇	—	4	2	no	—
61	C₄⋊C₄×C₆₇	—	4	2	no	—
62	C₄⋊C₄×C₆₇	—	4	2	no	—
63	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₆₇	D₈	1	1	yes	characteristic
65	C₄²×C₆₇	SD₁₆	1	1	yes	characteristic
66	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₂³×C₆₇	#102	1	1	yes	characteristic
68	Q₈×C₆₇	—	8	4	no	—
69	Q₈×C₆₇	—	8	4	no	—
70	C₂×C₄×C₆₇	—	2	2	no	—
71	C₂×C₄×C₆₇	—	4	4	no	—
72	C₂×C₄×C₆₇	—	8	4	no	—
73	C₂×C₄×C₆₇	#101	1	2	yes	—
74	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
75	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
76	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
77	C₂×C₄×C₆₇	#103	1	2	yes	—
78	C₂×C₄×C₆₇	SD₃₂	1	2	yes	—
79	C₂×C₄×C₆₇	Q₃₂	1	2	yes	—
80	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
83	C₈×C₆₇	—	2	1	no	—
84	C₈×C₆₇	—	2	1	no	—
85	C₈×C₆₇	#103	1	2	yes	—
86	C₈×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
87	C₈×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
88	C₆₇⋊C₈	—	8	2	no	—
89	C₂²×C₆₇	#332	1	2	yes	—
90	C₂²×C₆₇	#358	1	2	yes	—
91	C₂²×C₆₇	#361	1	2	yes	—
92	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
93	C₄×C₆₇	—	2	1	no	—
94	C₄×C₆₇	—	2	1	no	—
95	C₄×C₆₇	—	8	4	no	—
96	C₄×C₆₇	#332	1	2	yes	—
97	C₄×C₆₇	#358	1	2	yes	—
98	C₄×C₆₇	C₂×#101	1	1	yes	characteristic
99	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
100	C₄×C₆₇	C₂×SD₃₂	1	1	yes	characteristic
101	C₄×C₆₇	C₂×Q₃₂	1	1	yes	characteristic
102	C₂×#1365	—	67	1	no	maximal, Sylow
103	C₂×C₆₇	#1365	1	4	yes	complemented
104	C₂×C₆₇	C₂×#332	1	1	yes	characteristic
105	C₂×C₆₇	C₂×#358	1	1	yes	characteristic
106	C₂×C₆₇	C₂×#361	1	1	yes	characteristic
107	C₂×#500	D₆₇	1	1	yes	characteristic
108	C₂×C₄⋊C₁₆	—	67	1	no	—
109	C₂×#335	—	67	1	no	—
110	#1365	—	67	4	no	—
111	C₆₇	C₂×#1365	1	1	yes	characteristic, complemented, Sylow
112	C₂×C₄⋊Q₈	—	2	1	no	—
113	C₂²×C₁₆	—	134	1	no	—
114	#500	—	2	1	no	—
115	#500	—	2	1	no	—
116	#500	Dic₆₇	1	2	yes	—
117	#500	C₂×D₆₇	1	2	yes	—
118	C₂×#107	—	2	1	no	—
119	C₂×#107	Dic₆₇	1	2	yes	—
120	C₂×C₄⋊C₈	—	134	1	no	—
121	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
122	C₄⋊C₁₆	—	67	2	no	—
123	C₄⋊C₁₆	—	134	1	no	—
124	#335	—	67	2	no	—
125	#335	—	134	1	no	—
126	C₂²×Q₈	—	4	1	no	—
127	C₂²×C₈	—	268	1	no	—
128	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
129	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
130	C₄⋊Q₈	—	2	2	no	—
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	—	4	1	no	—
133	C₄⋊Q₈	—	4	1	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	4	1	no	—
136	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
137	C₂×C₁₆	—	268	2	no	—
138	C₂×C₁₆	—	134	2	no	—
139	#107	—	2	2	no	—
140	#107	—	2	2	no	—
141	#107	—	2	2	no	—
142	#107	C₂×Dic₆₇	1	4	yes	—
143	C₄⋊C₈	—	268	1	no	—
144	C₄⋊C₈	—	134	2	no	—
145	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
146	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
147	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
148	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
149	C₂×Q₈	—	4	2	no	—
150	C₂×Q₈	—	4	2	no	—
151	C₂×Q₈	—	8	4	no	—
152	C₂²×C₄	—	4	2	no	—
153	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
154	C₂²×C₄	#11761613	1	1	yes	characteristic
155	C₂²×C₄	#11761616	1	1	yes	characteristic
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	268	2	no	—
158	C₂×C₈	—	536	2	no	—
159	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
160	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
161	C₂×C₈	#11761615	1	2	yes	—
162	C₂×C₈	#11761618	1	2	yes	—
163	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
164	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄⋊C₄	—	4	2	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
170	C₄²	#11761613	1	1	yes	characteristic
171	C₄²	#11761616	1	1	yes	characteristic
172	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
173	C₁₆	—	268	2	no	—
174	C₂³	#14523964	1	1	yes	characteristic
175	Q₈	—	8	4	no	—
176	Q₈	—	8	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	8	4	no	—
180	C₂×C₄	#14523962	1	2	yes	—
181	C₂×C₄	#14523964	1	1	yes	characteristic
182	C₂×C₄	#14523964	1	1	yes	characteristic
183	C₂×C₄	#14523964	1	1	yes	characteristic
184	C₂×C₄	#14523967	1	2	yes	—
185	C₂×C₄	#14523986	1	2	yes	—
186	C₂×C₄	#14523989	1	2	yes	—
187	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
188	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
189	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
190	C₈	—	2	1	no	—
191	C₈	—	2	1	no	—
192	C₈	—	536	2	no	—
193	C₈	#14523967	1	2	yes	—
194	C₈	C₂×#11761615	1	1	yes	characteristic
195	C₈	C₂×#11761618	1	1	yes	characteristic
196	C₂²	#16863142	1	2	yes	—
197	C₂²	#16863200	1	2	yes	—
198	C₂²	#16863209	1	2	yes	—
199	C₂²	C₂×#14523964	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	8	4	no	—
203	C₄	#16863142	1	2	yes	—
204	C₄	#16863200	1	2	yes	—
205	C₄	C₂×#14523962	1	1	yes	characteristic
206	C₄	C₂×#14523967	1	1	yes	characteristic
207	C₄	C₂×#14523986	1	1	yes	characteristic
208	C₄	C₂×#14523989	1	1	yes	characteristic
209	C₂	#22558267	1	4	yes	direct product factor
210	C₂	C₂×#16863142	1	1	yes	characteristic
211	C₂	C₂×#16863200	1	1	yes	characteristic
212	C₂	C₂×#16863209	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14523964
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₂×#500×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	#102
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	C₂×#500	D₆₇
hypocenter	C₆₇	C₂×#1365
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
3. #22558267	C₂	direct (N × H)
103. C₂×C₆₇	#1365	semidirect
111. C₆₇	C₂×#1365	semidirect
209. C₂	#22558267	direct (N × H)

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
C₂×#16863149	2
#22558267	2
C₂×#500×C₆₇	2
C₂×#1365	67

Subgroups — C2×#22558267 #30767113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558267 #30767113