Subgroups — C₂×#22558191 #30765646

Order 17152.

346 conjugacy classes of subgroups.

The subgroup lattice has 232 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×#16863172	C₂	1	1	yes	maximal, characteristic
3	#22558191	C₂	1	4	yes	maximal, direct product factor
4	C₂×#446×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₆₇⋊C₁₆	—	2	1	no	—
6	C₂²×C₆₇⋊C₁₆	—	2	1	no	—
7	C₄⋊(C₆₇⋊C₁₆)	—	2	1	no	—
8	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
9	#16863172	—	2	1	no	—
10	#16863172	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
12	#446×C₆₇	—	2	1	no	—
13	#446×C₆₇	—	2	1	no	—
14	#446×C₆₇	C₄	1	2	yes	—
15	#446×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
17	C₂×C₄⋊C₈×C₆₇	C₄	1	1	yes	characteristic
18	C₂×#97×C₆₇	—	2	1	no	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₆₇⋊C₁₆	—	2	4	no	—
21	C₂×C₆₇⋊C₁₆	—	2	2	no	—
22	C₂×C₆₇⋊C₁₆	—	2	2	no	—
23	C₂×C₆₇⋊C₁₆	—	4	2	no	—
24	C₂²×Q₈×C₆₇	C₈	1	1	yes	characteristic
25	C₂²×C₈×C₆₇	—	2	1	no	—
26	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
28	C₄×Q₈×C₆₇	—	2	1	no	—
29	C₄×Q₈×C₆₇	—	2	1	no	—
30	C₄×Q₈×C₆₇	C₈	1	2	yes	—
31	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
33	C₂×C₄⋊C₄×C₆₇	C₈	1	1	yes	characteristic
34	C₂×C₄²×C₆₇	—	2	1	no	—
35	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₄⋊C₈×C₆₇	—	2	2	no	—
37	C₄⋊C₈×C₆₇	—	2	2	no	—
38	C₄⋊C₈×C₆₇	—	2	1	no	—
39	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
40	#97×C₆₇	—	2	2	no	—
41	#97×C₆₇	—	2	2	no	—
42	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
45	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
46	C₆₇⋊C₁₆	—	2	4	no	—
47	C₆₇⋊C₁₆	—	4	2	no	—
48	C₂×Q₈×C₆₇	—	2	1	no	—
49	C₂×Q₈×C₆₇	—	4	2	no	—
50	C₂×Q₈×C₆₇	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₆₇	—	2	1	no	—
52	C₂²×C₄×C₆₇	—	2	1	no	—
53	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
56	C₂×C₈×C₆₇	—	2	2	no	—
57	C₂×C₈×C₆₇	—	2	2	no	—
58	C₂×C₈×C₆₇	—	4	2	no	—
59	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
62	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
63	C₂×C₈×C₆₇	C₂×D₄	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	—	2	1	no	—
67	C₄⋊C₄×C₆₇	—	2	2	no	—
68	C₄⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
69	C₄²×C₆₇	—	2	2	no	—
70	C₄²×C₆₇	—	2	2	no	—
71	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
73	C₄²×C₆₇	M₁₆	1	1	yes	characteristic
74	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂³×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
76	Q₈×C₆₇	—	4	2	no	—
77	Q₈×C₆₇	—	4	2	no	—
78	C₂×C₄×C₆₇	—	2	2	no	—
79	C₂×C₄×C₆₇	—	2	2	no	—
80	C₂×C₄×C₆₇	—	2	2	no	—
81	C₂×C₄×C₆₇	—	2	2	no	—
82	C₂×C₄×C₆₇	—	4	2	no	—
83	C₂×C₄×C₆₇	—	4	2	no	—
84	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
85	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
86	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
87	C₂×C₄×C₆₇	#103	1	2	yes	—
88	C₂×C₄×C₆₇	#104	1	2	yes	—
89	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
92	C₈×C₆₇	—	2	1	no	—
93	C₈×C₆₇	—	2	1	no	—
94	C₈×C₆₇	—	4	2	no	—
95	C₈×C₆₇	#103	1	2	yes	—
96	C₈×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
97	C₈×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
98	C₂²×C₆₇	Q₈⋊C₈	1	2	yes	—
99	C₂²×C₆₇	#349	1	2	yes	—
100	C₂²×C₆₇	#350	1	2	yes	—
101	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
102	C₄×C₆₇	—	2	1	no	—
103	C₄×C₆₇	—	2	1	no	—
104	C₄×C₆₇	—	2	1	no	—
105	C₄×C₆₇	—	2	1	no	—
106	C₄×C₆₇	—	4	2	no	—
107	C₄×C₆₇	—	4	2	no	—
108	C₄×C₆₇	Q₈⋊C₈	1	2	yes	—
109	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
110	C₄×C₆₇	C₂×#104	1	1	yes	characteristic
111	C₂×#1338	—	67	1	no	maximal, Sylow
112	C₂×C₆₇	#1338	1	4	yes	complemented
113	C₂×C₆₇	C₂×Q₈⋊C₈	1	1	yes	characteristic
114	C₂×C₆₇	C₂×#349	1	1	yes	characteristic
115	C₂×C₆₇	C₂×#350	1	1	yes	characteristic
116	C₂×#446	D₆₇	1	1	yes	characteristic
117	C₂×C₄⋊C₁₆	—	67	1	no	—
118	C₂×#346	—	67	1	no	—
119	#1338	—	67	4	no	—
120	C₆₇	C₂×#1338	1	1	yes	characteristic, complemented, Sylow
121	C₂×C₄×Q₈	Dic₆₇	1	1	yes	characteristic
122	C₂²×C₁₆	—	134	1	no	—
123	C₂²×C₁₆	—	134	1	no	—
124	#446	—	2	1	no	—
125	#446	—	2	1	no	—
126	#446	Dic₆₇	1	2	yes	—
127	#446	C₂×D₆₇	1	2	yes	—
128	C₂×C₄⋊C₈	—	2	1	no	—
129	C₂×C₄⋊C₈	Dic₆₇	1	1	yes	characteristic
130	C₂×#97	—	2	1	no	—
131	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
132	C₄⋊C₁₆	—	67	2	no	—
133	C₄⋊C₁₆	—	134	1	no	—
134	#346	—	67	2	no	—
135	#346	—	134	1	no	—
136	C₂²×Q₈	C₆₇⋊C₈	1	1	yes	characteristic
137	C₂²×C₈	—	2	1	no	—
138	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
139	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
140	C₄×Q₈	—	2	1	no	—
141	C₄×Q₈	—	2	1	no	—
142	C₄×Q₈	C₆₇⋊C₈	1	2	yes	—
143	C₄×Q₈	C₂×Dic₆₇	1	2	yes	—
144	C₂×C₄⋊C₄	—	2	1	no	—
145	C₂×C₄⋊C₄	C₆₇⋊C₈	1	1	yes	characteristic
146	C₂×C₄²	—	2	1	no	—
147	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
148	C₂×C₁₆	—	268	2	no	—
149	C₂×C₁₆	—	134	2	no	—
150	C₂×C₁₆	—	134	4	no	—
151	C₂×C₁₆	—	134	2	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	2	1	no	—
155	C₄⋊C₈	C₂×Dic₆₇	1	2	yes	—
156	#97	—	2	2	no	—
157	#97	—	2	2	no	—
158	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
159	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
160	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
161	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
162	C₂×Q₈	—	2	1	no	—
163	C₂×Q₈	—	4	2	no	—
164	C₂×Q₈	C₂×C₆₇⋊C₈	1	2	yes	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
169	C₂²×C₄	#11761612	1	1	yes	characteristic
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
174	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
175	C₂×C₈	#11761615	1	2	yes	—
176	C₂×C₈	#11761618	1	2	yes	—
177	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
178	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	C₂×C₆₇⋊C₈	1	2	yes	—
183	C₄²	—	2	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
186	C₄²	C₂×C₆₇⋊C₈	1	1	yes	characteristic
187	C₄²	#11761612	1	1	yes	characteristic
188	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
189	C₁₆	—	268	2	no	—
190	C₁₆	—	134	4	no	—
191	C₂³	C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
192	Q₈	—	4	2	no	—
193	Q₈	—	4	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
201	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
202	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
203	C₂×C₄	#14523967	1	2	yes	—
204	C₂×C₄	#14523970	1	2	yes	—
205	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
206	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
207	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
208	C₈	—	2	1	no	—
209	C₈	—	2	1	no	—
210	C₈	—	4	2	no	—
211	C₈	#14523967	1	2	yes	—
212	C₈	C₂×#11761615	1	1	yes	characteristic
213	C₈	C₂×#11761618	1	1	yes	characteristic
214	C₂²	Q₈⋊(C₆₇⋊C₈)	1	2	yes	—
215	C₂²	#16863178	1	2	yes	—
216	C₂²	#16863180	1	2	yes	—
217	C₂²	C₂×C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	4	2	no	—
223	C₄	—	4	2	no	—
224	C₄	Q₈⋊(C₆₇⋊C₈)	1	2	yes	—
225	C₄	C₂×#14523967	1	1	yes	characteristic
226	C₄	C₂×#14523970	1	1	yes	characteristic
227	C₂	#22558191	1	4	yes	direct product factor
228	C₂	C₂×Q₈⋊(C₆₇⋊C₈)	1	1	yes	characteristic
229	C₂	C₂×#16863178	1	1	yes	characteristic
230	C₂	C₂×#16863180	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₂×#446×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²⋊C₈
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	C₂×#446	D₆₇
hypocenter	C₆₇	C₂×#1338
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
3. #22558191	C₂	direct (N × H)
112. C₂×C₆₇	#1338	semidirect
120. C₆₇	C₂×#1338	semidirect
227. C₂	#22558191	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×#16863172	2
#22558191	2
C₂×#446×C₆₇	2
C₂×#1338	67

Subgroups — C2×#22558191 #30765646

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558191 #30765646