Subgroups — C₆₇×#21832 #30701083

Order 17152.

914 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#544×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2921	C₂	1	2	yes	maximal
3	C₆₇×#1998	C₂	1	4	yes	maximal, complemented
4	C₆₇×#1997	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1822	C₂	1	2	yes	maximal
6	C₆₇×#1821	C₂	1	2	yes	maximal
7	#544×C₆₇	—	2	8	no	—
8	C₂×C₄⋊Q₈×C₆₇	C₄	1	2	yes	—
9	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
10	C₂×#125×C₆₇	C₄	1	2	yes	—
11	C₂×#125×C₆₇	C₂²	1	2	yes	—
12	C₂×C₂²⋊Q₈×C₆₇	C₄	1	4	yes	—
13	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
16	#433×C₆₇	—	2	8	no	—
17	#423×C₆₇	C₂²	1	4	yes	—
18	#407×C₆₇	C₂²	1	4	yes	—
19	C₄×M₁₆×C₆₇	C₂²	1	4	yes	—
20	#343×C₆₇	C₂²	1	4	yes	—
21	#343×C₆₇	C₂²	1	8	yes	—
22	#343×C₆₇	C₂²	1	4	yes	—
23	C₂²×Q₈×C₆₇	C₂×C₄	1	4	yes	—
24	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
27	C₄⋊Q₈×C₆₇	—	2	8	no	—
28	C₄⋊Q₈×C₆₇	—	2	4	no	—
29	C₄⋊Q₈×C₆₇	—	2	4	no	—
30	#125×C₆₇	—	2	8	no	—
31	#125×C₆₇	—	2	4	no	—
32	#125×C₆₇	—	2	4	no	—
33	C₂²⋊Q₈×C₆₇	—	2	16	no	—
34	#117×C₆₇	—	2	4	no	—
35	#117×C₆₇	D₄	1	8	yes	—
36	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
37	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
38	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
39	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
40	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
41	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
44	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
50	C₄⋊C₈×C₆₇	—	2	8	no	—
51	C₂²⋊C₈×C₆₇	—	2	8	no	—
52	#97×C₆₇	—	2	4	no	—
53	C₄×C₈×C₆₇	—	2	4	no	—
54	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×Q₈×C₆₇	—	2	8	no	—
56	C₂×Q₈×C₆₇	—	8	4	no	—
57	C₂²×C₄×C₆₇	—	2	4	no	—
58	C₂²×C₄×C₆₇	—	2	4	no	—
59	C₂²×C₄×C₆₇	—	8	2	no	—
60	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
67	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
68	M₁₆×C₆₇	—	2	16	no	—
69	C₂×C₈×C₆₇	—	2	8	no	—
70	C₄⋊C₄×C₆₇	—	2	4	no	—
71	C₄⋊C₄×C₆₇	—	2	8	no	—
72	C₄⋊C₄×C₆₇	—	2	8	no	—
73	C₄⋊C₄×C₆₇	—	2	8	no	—
74	C₄⋊C₄×C₆₇	—	2	2	no	—
75	C₄⋊C₄×C₆₇	—	2	2	no	—
76	C₄⋊C₄×C₆₇	—	2	2	no	—
77	C₄⋊C₄×C₆₇	—	2	2	no	—
78	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
79	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
80	C₂²⋊C₄×C₆₇	—	2	4	no	—
81	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
82	C₄²×C₆₇	—	2	4	no	—
83	C₄²×C₆₇	C₂×D₄	1	4	yes	—
84	C₄²×C₆₇	C₂×D₄	1	4	yes	—
85	C₂³×C₆₇	—	2	2	no	—
86	C₂³×C₆₇	—	4	1	no	—
87	C₂³×C₆₇	—	4	1	no	—
88	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
89	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
90	Q₈×C₆₇	—	8	8	no	—
91	C₂×C₄×C₆₇	—	2	2	no	—
92	C₂×C₄×C₆₇	—	2	8	no	—
93	C₂×C₄×C₆₇	—	2	2	no	—
94	C₂×C₄×C₆₇	—	2	2	no	—
95	C₂×C₄×C₆₇	—	2	4	no	—
96	C₂×C₄×C₆₇	—	4	2	no	—
97	C₂×C₄×C₆₇	—	4	2	no	—
98	C₂×C₄×C₆₇	—	4	4	no	—
99	C₂×C₄×C₆₇	—	4	2	no	—
100	C₂×C₄×C₆₇	—	4	2	no	—
101	C₂×C₄×C₆₇	—	8	4	no	—
102	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
103	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
104	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
105	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
106	C₂×C₄×C₆₇	2₋¹⁺⁴	1	4	yes	—
107	C₈×C₆₇	—	4	8	no	—
108	C₂²×C₆₇	—	2	1	no	—
109	C₂²×C₆₇	—	2	1	no	—
110	C₂²×C₆₇	—	2	2	no	—
111	C₂²×C₆₇	—	2	2	no	—
112	C₂²×C₆₇	—	2	2	no	—
113	C₂²×C₆₇	—	4	4	no	—
114	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
115	C₂²×C₆₇	#519	1	2	yes	—
116	C₄×C₆₇	—	2	2	no	—
117	C₄×C₆₇	—	2	4	no	—
118	C₄×C₆₇	—	2	4	no	—
119	C₄×C₆₇	—	4	4	no	—
120	C₄×C₆₇	—	4	2	no	—
121	C₄×C₆₇	—	4	2	no	—
122	C₄×C₆₇	#536	1	4	yes	—
123	#21832	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂×C₆₇	—	2	2	no	—
125	C₂×C₆₇	—	4	2	no	—
126	C₂×C₆₇	#2310	1	1	yes	characteristic
127	C₂×C₆₇	#2888	1	1	yes	characteristic
128	C₂×C₆₇	#2889	1	1	yes	characteristic
129	C₂×#544	C₂×C₆₇	1	1	yes	characteristic
130	#2921	C₂×C₆₇	1	2	yes	—
131	#1998	C₂×C₆₇	1	4	yes	complemented
132	#1997	C₂×C₆₇	1	4	yes	complemented
133	#1822	C₂×C₆₇	1	2	yes	—
134	#1821	C₂×C₆₇	1	2	yes	—
135	C₆₇	#21832	1	1	yes	characteristic, direct product factor, Sylow
136	#544	—	2	8	no	—
137	C₂×C₄⋊Q₈	C₄×C₆₇	1	2	yes	—
138	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
139	C₂×#125	C₄×C₆₇	1	2	yes	—
140	C₂×#125	C₂²×C₆₇	1	2	yes	—
141	C₂×C₂²⋊Q₈	C₄×C₆₇	1	4	yes	—
142	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
143	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
144	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
145	#433	—	2	8	no	—
146	#423	C₂²×C₆₇	1	4	yes	—
147	#407	C₂²×C₆₇	1	4	yes	—
148	C₄×M₁₆	C₂²×C₆₇	1	4	yes	—
149	#343	C₂²×C₆₇	1	8	yes	—
150	#343	C₂²×C₆₇	1	4	yes	—
151	#343	C₂²×C₆₇	1	4	yes	—
152	C₂²×Q₈	C₂×C₄×C₆₇	1	4	yes	—
153	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
154	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
155	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
156	C₄⋊Q₈	—	2	8	no	—
157	C₄⋊Q₈	—	2	4	no	—
158	C₄⋊Q₈	—	2	4	no	—
159	#125	—	2	8	no	—
160	#125	—	2	4	no	—
161	#125	—	2	4	no	—
162	C₂²⋊Q₈	—	2	16	no	—
163	#117	—	2	4	no	—
164	#117	D₄×C₆₇	1	8	yes	—
165	C₂×C₄⋊C₄	—	2	4	no	—
166	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
167	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
168	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
169	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
170	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
171	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
172	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
173	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
174	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
175	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
176	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
177	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
178	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
179	C₄⋊C₈	—	2	8	no	—
180	C₂²⋊C₈	—	2	8	no	—
181	#97	—	2	4	no	—
182	C₄×C₈	—	2	4	no	—
183	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
184	C₂×Q₈	—	2	8	no	—
185	C₂×Q₈	—	8	4	no	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	8	2	no	—
189	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
190	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
191	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
192	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
193	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
194	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
195	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
196	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
197	M₁₆	—	2	16	no	—
198	C₂×C₈	—	2	8	no	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	2	8	no	—
201	C₄⋊C₄	—	2	8	no	—
202	C₄⋊C₄	—	2	8	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
208	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
209	C₂²⋊C₄	—	2	4	no	—
210	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
211	C₄²	—	2	4	no	—
212	C₄²	C₂×D₄×C₆₇	1	4	yes	—
213	C₄²	C₂×D₄×C₆₇	1	4	yes	—
214	C₂³	—	2	2	no	—
215	C₂³	—	4	1	no	—
216	C₂³	—	4	1	no	—
217	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
218	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
219	Q₈	—	8	8	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	8	4	no	—
231	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
232	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
233	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
234	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
235	C₂×C₄	2₋¹⁺⁴×C₆₇	1	4	yes	—
236	C₈	—	4	8	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	4	4	no	—
243	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
244	C₂²	#519×C₆₇	1	2	yes	—
245	C₄	—	2	2	no	—
246	C₄	—	2	4	no	—
247	C₄	—	2	4	no	—
248	C₄	—	4	4	no	—
249	C₄	—	4	2	no	—
250	C₄	—	4	2	no	—
251	C₄	#536×C₆₇	1	4	yes	—
252	C₂	—	2	2	no	—
253	C₂	—	4	2	no	—
254	C₂	C₆₇×#2310	1	1	yes	characteristic
255	C₂	C₆₇×#2888	1	1	yes	characteristic
256	C₂	C₆₇×#2889	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	C₂⁴×C₆₇	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#1998	C₂	semidirect
4. C₆₇×#1997	C₂	semidirect
123. #21832	C₆₇	direct (N × H)
131. #1998	C₂×C₆₇	semidirect
132. #1997	C₂×C₆₇	semidirect
135. C₆₇	#21832	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×#544×C₆₇	2
C₆₇×#2921	2
C₆₇×#1998	2
C₆₇×#1997	2
C₆₇×#1822	2
C₆₇×#1821	2
#21832	67

Subgroups — C67×#21832 #30701083

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21832 #30701083