Subgroups — C₆₇×#22536 #30701787

Order 17152.

1314 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#550×C₆₇	C₂	1	1	yes	maximal, characteristic
2	Q₈×M₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1965	C₂	1	6	yes	maximal, complemented
4	C₆₇×#1800	C₂	1	3	yes	maximal, complemented
5	C₄×#100×C₆₇	C₂	1	3	yes	maximal, complemented
6	C₂²×#41×C₆₇	C₄	1	2	yes	—
7	#550×C₆₇	—	2	2	no	—
8	#550×C₆₇	—	2	6	no	—
9	C₂×#127×C₆₇	C₂²	1	3	yes	—
10	C₂×#121×C₆₇	C₄	1	6	yes	—
11	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₄×C₆₇	C₂²	1	3	yes	—
13	#446×C₆₇	—	2	6	no	—
14	C₈×Q₈×C₆₇	—	2	2	no	—
15	#423×C₆₇	C₂²	1	6	yes	—
16	C₂×#100×C₆₇	C₂²	1	6	yes	—
17	C₂×#100×C₆₇	C₂²	1	6	yes	—
18	C₄×M₁₆×C₆₇	C₂²	1	6	yes	—
19	#343×C₆₇	C₂²	1	1	yes	characteristic
20	#343×C₆₇	C₂²	1	3	yes	—
21	C₂×#41×C₆₇	—	2	6	no	—
22	C₂×#41×C₆₇	—	2	2	no	—
23	C₂×#41×C₆₇	—	4	8	no	—
24	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×D₄×C₆₇	C₂×C₄	1	6	yes	—
26	C₂²×D₄×C₆₇	C₂×C₄	1	6	yes	—
27	C₂²×D₄×C₆₇	C₂³	1	3	yes	—
28	C₂³×C₄×C₆₇	C₂×C₄	1	6	yes	—
29	C₂×M₁₆×C₆₇	C₂³	1	6	yes	—
30	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
31	#127×C₆₇	—	2	6	no	—
32	#127×C₆₇	—	2	6	no	—
33	#121×C₆₇	—	2	12	no	—
34	#121×C₆₇	—	2	12	no	—
35	C₄×Q₈×C₆₇	D₄	1	2	yes	—
36	C₄×Q₈×C₆₇	D₄	1	6	yes	—
37	C₄×D₄×C₆₇	—	2	6	no	—
38	C₄×D₄×C₆₇	—	2	6	no	—
39	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	3	yes	—
40	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	6	yes	—
42	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
43	C₄⋊C₈×C₆₇	—	2	6	no	—
44	C₄⋊C₈×C₆₇	—	2	6	no	—
45	#100×C₆₇	—	2	12	no	—
46	#100×C₆₇	—	4	6	no	—
47	#97×C₆₇	—	2	6	no	—
48	C₄×C₈×C₆₇	—	2	6	no	—
49	C₂⁴×C₆₇	C₂²×C₄	1	6	yes	—
50	#41×C₆₇	—	4	24	no	—
51	#41×C₆₇	—	4	8	no	—
52	C₂×Q₈×C₆₇	—	2	4	no	—
53	C₂×Q₈×C₆₇	C₂²⋊C₄	1	3	yes	—
54	C₂×Q₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×D₄×C₆₇	—	2	6	no	—
56	C₂×D₄×C₆₇	—	2	6	no	—
57	C₂×D₄×C₆₇	—	2	12	no	—
58	C₂×D₄×C₆₇	—	2	6	no	—
59	C₂×D₄×C₆₇	—	2	6	no	—
60	C₂×D₄×C₆₇	—	2	3	no	—
61	C₂×D₄×C₆₇	—	2	3	no	—
62	C₂×D₄×C₆₇	—	4	12	no	—
63	C₂×D₄×C₆₇	—	8	6	no	—
64	C₂²×C₄×C₆₇	—	2	6	no	—
65	C₂²×C₄×C₆₇	—	2	6	no	—
66	C₂²×C₄×C₆₇	—	4	12	no	—
67	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
68	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
70	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
71	M₁₆×C₆₇	—	2	2	no	—
72	M₁₆×C₆₇	—	2	6	no	—
73	M₁₆×C₆₇	—	2	6	no	—
74	M₁₆×C₆₇	#41	1	4	yes	complemented
75	C₂×C₈×C₆₇	—	2	2	no	—
76	C₂×C₈×C₆₇	—	2	6	no	—
77	C₄⋊C₄×C₆₇	—	2	2	no	—
78	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	6	yes	—
79	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	6	yes	—
80	C₂²⋊C₄×C₆₇	—	2	12	no	—
81	C₄²×C₆₇	C₂×D₄	1	6	yes	—
82	C₄²×C₆₇	C₂×D₄	1	6	yes	—
83	C₂³×C₆₇	—	2	6	no	—
84	C₂³×C₆₇	—	2	6	no	—
85	C₂³×C₆₇	—	2	6	no	—
86	C₂³×C₆₇	—	4	6	no	—
87	C₂³×C₆₇	—	4	6	no	—
88	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
89	Q₈×C₆₇	—	2	6	no	—
90	Q₈×C₆₇	#100	1	4	yes	complemented
91	D₄×C₆₇	—	4	12	no	—
92	D₄×C₆₇	—	4	6	no	—
93	D₄×C₆₇	—	4	6	no	—
94	D₄×C₆₇	—	4	12	no	—
95	D₄×C₆₇	—	8	12	no	—
96	C₂×C₄×C₆₇	—	2	6	no	—
97	C₂×C₄×C₆₇	—	2	2	no	—
98	C₂×C₄×C₆₇	—	4	3	no	—
99	C₂×C₄×C₆₇	—	4	12	no	—
100	C₂×C₄×C₆₇	—	4	12	no	—
101	C₂×C₄×C₆₇	—	4	2	no	—
102	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	6	yes	—
103	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
104	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
105	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
106	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
107	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
108	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
109	C₈×C₆₇	—	2	4	no	—
110	C₈×C₆₇	—	4	6	no	—
111	C₂²×C₆₇	—	2	1	no	—
112	C₂²×C₆₇	—	2	1	no	—
113	C₂²×C₆₇	—	4	6	no	—
114	C₂²×C₆₇	—	4	6	no	—
115	C₂²×C₆₇	—	4	6	no	—
116	C₂²×C₆₇	—	8	6	no	—
117	C₂²×C₆₇	—	8	6	no	—
118	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
119	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
120	C₂²×C₆₇	#520	1	1	yes	characteristic
121	C₄×C₆₇	—	2	3	no	—
122	C₄×C₆₇	—	2	6	no	—
123	C₄×C₆₇	—	4	2	no	—
124	C₄×C₆₇	C₂×#100	1	6	yes	—
125	C₄×C₆₇	#548	1	2	yes	—
126	C₄×C₆₇	#550	1	2	yes	—
127	#22536	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂×C₆₇	—	2	1	no	—
129	C₂×C₆₇	—	2	1	no	—
130	C₂×C₆₇	—	8	6	no	—
131	C₂×C₆₇	#2341	1	1	yes	characteristic
132	C₂×C₆₇	C₂²×#100	1	1	yes	characteristic
133	C₂×C₆₇	#2888	1	1	yes	characteristic
134	C₂×#550	C₂×C₆₇	1	1	yes	characteristic
135	Q₈×M₁₆	C₂×C₆₇	1	2	yes	complemented
136	#1965	C₂×C₆₇	1	6	yes	complemented
137	#1800	C₂×C₆₇	1	3	yes	complemented
138	C₄×#100	C₂×C₆₇	1	3	yes	complemented
139	C₆₇	#22536	1	1	yes	characteristic, direct product factor, Sylow
140	C₂²×#41	C₄×C₆₇	1	2	yes	—
141	#550	—	2	2	no	—
142	#550	—	2	6	no	—
143	C₂×#127	C₂²×C₆₇	1	3	yes	—
144	C₂×#121	C₄×C₆₇	1	6	yes	—
145	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
146	C₂×C₄×D₄	C₂²×C₆₇	1	3	yes	—
147	#446	—	2	6	no	—
148	C₈×Q₈	—	2	2	no	—
149	#423	C₂²×C₆₇	1	6	yes	—
150	C₂×#100	C₂²×C₆₇	1	6	yes	—
151	C₂×#100	C₂²×C₆₇	1	6	yes	—
152	C₄×M₁₆	C₂²×C₆₇	1	6	yes	—
153	#343	C₂²×C₆₇	1	3	yes	—
154	#343	C₂²×C₆₇	1	1	yes	characteristic
155	C₂×#41	—	2	6	no	—
156	C₂×#41	—	2	2	no	—
157	C₂×#41	—	4	8	no	—
158	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
159	C₂²×D₄	C₂×C₄×C₆₇	1	6	yes	—
160	C₂²×D₄	C₂×C₄×C₆₇	1	6	yes	—
161	C₂²×D₄	C₂³×C₆₇	1	3	yes	—
162	C₂³×C₄	C₂×C₄×C₆₇	1	6	yes	—
163	C₂×M₁₆	C₂³×C₆₇	1	6	yes	—
164	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
165	#127	—	2	6	no	—
166	#127	—	2	6	no	—
167	#121	—	2	12	no	—
168	#121	—	2	12	no	—
169	C₄×Q₈	D₄×C₆₇	1	2	yes	—
170	C₄×Q₈	D₄×C₆₇	1	6	yes	—
171	C₄×D₄	—	2	6	no	—
172	C₄×D₄	—	2	6	no	—
173	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	3	yes	—
174	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
175	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	6	yes	—
176	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
177	C₄⋊C₈	—	2	6	no	—
178	C₄⋊C₈	—	2	6	no	—
179	#100	—	2	12	no	—
180	#100	—	4	6	no	—
181	#97	—	2	6	no	—
182	C₄×C₈	—	2	6	no	—
183	C₂⁴	C₂²×C₄×C₆₇	1	6	yes	—
184	#41	—	4	24	no	—
185	#41	—	4	8	no	—
186	C₂×Q₈	—	2	4	no	—
187	C₂×Q₈	C₂²⋊C₄×C₆₇	1	3	yes	—
188	C₂×Q₈	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
189	C₂×D₄	—	2	6	no	—
190	C₂×D₄	—	2	6	no	—
191	C₂×D₄	—	2	12	no	—
192	C₂×D₄	—	2	6	no	—
193	C₂×D₄	—	2	6	no	—
194	C₂×D₄	—	2	3	no	—
195	C₂×D₄	—	2	3	no	—
196	C₂×D₄	—	4	12	no	—
197	C₂×D₄	—	8	6	no	—
198	C₂²×C₄	—	2	6	no	—
199	C₂²×C₄	—	2	6	no	—
200	C₂²×C₄	—	4	12	no	—
201	C₂²×C₄	C₂²×C₄×C₆₇	1	3	yes	—
202	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
203	C₂²×C₄	C₂²×C₄×C₆₇	1	3	yes	—
204	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
205	M₁₆	—	2	2	no	—
206	M₁₆	—	2	6	no	—
207	M₁₆	—	2	6	no	—
208	M₁₆	#41×C₆₇	1	4	yes	complemented
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	6	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	6	yes	—
213	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	6	yes	—
214	C₂²⋊C₄	—	2	12	no	—
215	C₄²	C₂×D₄×C₆₇	1	6	yes	—
216	C₄²	C₂×D₄×C₆₇	1	6	yes	—
217	C₂³	—	2	6	no	—
218	C₂³	—	2	6	no	—
219	C₂³	—	2	6	no	—
220	C₂³	—	4	6	no	—
221	C₂³	—	4	6	no	—
222	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
223	Q₈	—	2	6	no	—
224	Q₈	#100×C₆₇	1	4	yes	complemented
225	D₄	—	4	12	no	—
226	D₄	—	4	6	no	—
227	D₄	—	4	6	no	—
228	D₄	—	4	12	no	—
229	D₄	—	8	12	no	—
230	C₂×C₄	—	2	6	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	3	no	—
233	C₂×C₄	—	4	12	no	—
234	C₂×C₄	—	4	12	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	6	yes	—
237	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	3	yes	—
238	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	3	yes	—
239	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
240	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
241	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
242	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
243	C₈	—	2	4	no	—
244	C₈	—	4	6	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	4	6	no	—
248	C₂²	—	4	6	no	—
249	C₂²	—	4	6	no	—
250	C₂²	—	8	6	no	—
251	C₂²	—	8	6	no	—
252	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
253	C₂²	C₄×#41×C₆₇	1	1	yes	characteristic
254	C₂²	#520×C₆₇	1	1	yes	characteristic
255	C₄	—	2	3	no	—
256	C₄	—	2	6	no	—
257	C₄	—	4	2	no	—
258	C₄	C₂×#100×C₆₇	1	6	yes	—
259	C₄	#548×C₆₇	1	2	yes	—
260	C₄	#550×C₆₇	1	2	yes	—
261	C₂	—	2	1	no	—
262	C₂	—	2	1	no	—
263	C₂	—	8	6	no	—
264	C₂	C₆₇×#2341	1	1	yes	characteristic
265	C₂	C₂²×#100×C₆₇	1	1	yes	characteristic
266	C₂	C₆₇×#2888	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#550×C₆₇	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. Q₈×M₁₆×C₆₇	C₂	semidirect
3. C₆₇×#1965	C₂	semidirect
4. C₆₇×#1800	C₂	semidirect
5. C₄×#100×C₆₇	C₂	semidirect
74. M₁₆×C₆₇	#41	semidirect
90. Q₈×C₆₇	#100	semidirect
127. #22536	C₆₇	direct (N × H)
135. Q₈×M₁₆	C₂×C₆₇	semidirect
136. #1965	C₂×C₆₇	semidirect
137. #1800	C₂×C₆₇	semidirect
138. C₄×#100	C₂×C₆₇	semidirect
139. C₆₇	#22536	direct (N × H)
208. M₁₆	#41×C₆₇	semidirect
224. Q₈	#100×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#550×C₆₇	2
Q₈×M₁₆×C₆₇	2
C₆₇×#1965	2
C₆₇×#1800	2
C₄×#100×C₆₇	2
#22536	67

Subgroups — C67×#22536 #30701787

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22536 #30701787