Group #30772759 — subgroups

Subgroups — #30772759

Order 17152.

470 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568910	C₂	1	1	yes	maximal, characteristic
2	#22561830	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16863177	C₂	1	2	yes	maximal, complemented
4	#22558296	C₂	1	2	yes	maximal, complemented
5	C₆₇×#3407	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#11761611	C₂²	1	1	yes	characteristic
7	#16863903	C₄	1	1	yes	characteristic
8	C₂×#14523983	C₂²	1	2	yes	—
9	#16863472	—	2	2	no	—
10	#16863467	—	2	2	no	—
11	#16863424	C₄	1	1	yes	characteristic
12	#16863363	C₄	1	1	yes	characteristic
13	#16863359	C₄	1	1	yes	characteristic
14	#16863351	C₂²	1	1	yes	characteristic
15	#16863178	—	2	2	no	—
16	#16863177	—	2	4	no	—
17	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
18	#504×C₆₇	—	2	4	no	—
19	C₂×M₃₂×C₆₇	—	2	2	no	—
20	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
21	C₂²×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
23	C₂×#11761611	—	2	2	no	—
24	C₂×#11761611	—	2	1	no	—
25	C₂×#11761611	—	2	1	no	—
26	C₂×#11761611	—	2	2	no	—
27	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	#14524024	C₈	1	2	yes	—
29	#14524019	—	2	1	no	—
30	#14524014	—	2	1	no	—
31	#14524011	C₈	1	2	yes	—
32	C₄×C₂²⋊D₆₇	—	2	2	no	—
33	#14524002	C₂×C₄	1	1	yes	characteristic
34	#14523983	—	2	2	no	—
35	#14523983	—	2	2	no	—
36	C₂×C₆₇⋊C₁₆	—	2	2	no	—
37	Dic₆₇⋊C₈	—	2	2	no	—
38	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
39	(C₂×D₆₇)⋊C₈	—	2	2	no	—
40	#14523953	—	2	1	no	—
41	C₈×Dic₆₇	—	2	1	no	—
42	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
43	#131×C₆₇	—	2	2	no	—
44	#131×C₆₇	—	2	2	no	—
45	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
47	C₂²×C₈×C₆₇	D₄	1	2	yes	—
48	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
49	M₃₂×C₆₇	—	2	4	no	—
50	M₃₂×C₆₇	—	2	2	no	—
51	M₃₂×C₆₇	D₄	1	4	yes	complemented
52	C₂×C₁₆×C₆₇	—	2	4	no	—
53	C₂×C₁₆×C₆₇	—	2	2	no	—
54	C₂³×D₆₇	C₂×C₈	1	1	yes	characteristic
55	C₂×C₂²⋊D₆₇	—	2	2	no	—
56	C₂²×Dic₆₇	C₂×C₈	1	1	yes	characteristic
57	C₂×C₄×D₆₇	—	2	1	no	—
58	C₂×C₄×D₆₇	—	2	2	no	—
59	C₂×C₄×D₆₇	—	2	1	no	—
60	C₂×C₄×D₆₇	—	2	2	no	—
61	#11761612	—	2	2	no	—
62	#11761611	—	4	1	no	—
63	#11761611	—	4	2	no	—
64	#11761611	—	4	1	no	—
65	C₂×C₆₇⋊C₈	—	2	1	no	—
66	C₂×C₆₇⋊C₈	—	2	1	no	—
67	C₂×C₆₇⋊C₈	—	2	2	no	—
68	C₄⋊Dic₆₇	—	2	1	no	—
69	Dic₆₇⋊C₄	—	2	2	no	—
70	C₂²⋊Dic₆₇	—	2	2	no	—
71	C₂²⋊Dic₆₇	C₂×C₈	1	2	yes	—
72	(C₂×D₆₇)⋊C₄	—	2	2	no	—
73	C₄×Dic₆₇	—	2	1	no	—
74	C₆₇⋊C₁₆	—	4	2	no	—
75	#41×C₆₇	—	2	2	no	—
76	#41×C₆₇	—	2	2	no	—
77	C₂×Q₈×C₆₇	M₁₆	1	1	yes	characteristic
78	C₂×D₄×C₆₇	C₂×C₈	1	2	yes	—
79	C₂×D₄×C₆₇	M₁₆	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	M₁₆×C₆₇	—	2	2	no	—
83	M₁₆×C₆₇	—	2	4	no	—
84	C₂×C₈×C₆₇	—	2	2	no	—
85	C₂×C₈×C₆₇	—	2	2	no	—
86	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
87	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₆₇	#41	1	2	yes	—
91	C₁₆×C₆₇	—	2	4	no	—
92	C₂³×C₆₇	C₂²⋊C₈	1	2	yes	—
93	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
94	C₂²×D₆₇	—	2	1	no	—
95	C₂²×D₆₇	—	2	2	no	—
96	C₂²×D₆₇	—	8	1	no	—
97	Q₈×C₆₇	—	2	2	no	—
98	D₄×C₆₇	—	2	4	no	—
99	D₄×C₆₇	—	2	2	no	—
100	C₂×C₄×C₆₇	—	2	2	no	—
101	C₂×C₄×C₆₇	—	2	2	no	—
102	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
103	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
106	C₂×C₄×C₆₇	#131	1	2	yes	—
107	C₂²⋊D₆₇	—	8	2	no	—
108	C₂×Dic₆₇	—	2	2	no	—
109	C₂×Dic₆₇	—	2	1	no	—
110	C₂×Dic₆₇	—	2	2	no	—
111	C₄×D₆₇	—	4	1	no	—
112	C₄×D₆₇	—	4	1	no	—
113	C₄×D₆₇	—	4	2	no	—
114	C₈×C₆₇	—	2	2	no	—
115	C₈×C₆₇	#120	1	1	yes	characteristic
116	C₈×C₆₇	#121	1	2	yes	—
117	C₈×C₆₇	#123	1	1	yes	characteristic
118	C₆₇⋊C₈	—	4	1	no	—
119	C₆₇⋊C₈	—	4	1	no	—
120	C₂²×C₆₇	—	2	2	no	—
121	C₂²×C₆₇	—	4	1	no	—
122	C₂²×C₆₇	—	4	2	no	—
123	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
124	C₂²×C₆₇	C₈×D₄	1	2	yes	—
125	C₂×D₆₇	—	4	1	no	—
126	C₂×D₆₇	—	4	1	no	—
127	C₂×D₆₇	—	4	1	no	—
128	C₂×D₆₇	—	8	2	no	—
129	C₄×C₆₇	—	2	2	no	—
130	C₄×C₆₇	#386	1	1	yes	characteristic
131	C₄×C₆₇	#408	1	1	yes	characteristic
132	C₄×C₆₇	#435	1	2	yes	—
133	Dic₆₇	—	4	2	no	—
134	Dic₆₇	—	4	1	no	—
135	#12989	—	67	1	no	maximal, Sylow
136	C₂×C₆₇	—	2	1	no	—
137	C₂×C₆₇	—	2	2	no	—
138	C₂×C₆₇	—	4	2	no	—
139	C₂×C₆₇	#1869	1	1	yes	characteristic
140	D₆₇	—	8	1	no	—
141	#3407	D₆₇	1	1	yes	characteristic, complemented
142	#2930	—	67	1	no	—
143	#2115	—	67	1	no	—
144	C₂×#349	—	67	2	no	—
145	#1378	—	67	2	no	—
146	C₆₇	#12989	1	1	yes	characteristic, complemented, Sylow
147	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
148	C₂²×M₁₆	—	67	1	no	—
149	#525	—	67	1	no	—
150	#504	—	2	4	no	—
151	C₂×M₃₂	—	2	2	no	—
152	C₂×M₃₂	—	67	2	no	—
153	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
154	#436	—	134	2	no	—
155	#435	—	134	2	no	—
156	#424	—	67	1	no	—
157	#408	—	67	1	no	—
158	#407	—	67	1	no	—
159	#405	—	67	1	no	—
160	#349	—	134	2	no	—
161	#349	—	134	4	no	—
162	C₂×#41	C₄×D₆₇	1	1	yes	characteristic
163	C₂³×C₄	—	67	1	no	—
164	#131	—	2	2	no	—
165	#131	—	2	2	no	—
166	C₂×M₁₆	—	67	1	no	—
167	C₂×M₁₆	—	134	2	no	—
168	C₂×M₁₆	—	134	1	no	—
169	C₂×M₁₆	—	134	1	no	—
170	C₂×M₁₆	—	134	2	no	—
171	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
172	C₂×M₁₆	D₂₆₈	1	2	yes	—
173	C₂²×C₈	—	67	1	no	—
174	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
175	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
176	#123	—	67	2	no	—
177	C₂²⋊Q₈	—	134	1	no	—
178	#121	—	134	1	no	—
179	#120	—	67	2	no	—
180	C₄×D₄	—	134	2	no	—
181	#117	—	67	1	no	—
182	M₃₂	—	2	4	no	—
183	M₃₂	—	2	2	no	—
184	M₃₂	—	134	2	no	—
185	M₃₂	—	134	2	no	—
186	M₃₂	C₂²⋊D₆₇	1	4	yes	complemented
187	C₂×C₁₆	—	2	4	no	—
188	C₂×C₁₆	—	2	2	no	—
189	C₂×C₁₆	—	134	2	no	—
190	C₄⋊C₈	—	134	2	no	—
191	C₂²⋊C₈	—	134	2	no	—
192	C₂²⋊C₈	—	134	2	no	—
193	#97	—	134	1	no	—
194	C₄×C₈	—	134	1	no	—
195	C₂⁴	—	67	1	no	—
196	#41	—	2	2	no	—
197	#41	—	2	2	no	—
198	C₂×Q₈	#11761611	1	1	yes	characteristic
199	C₂×D₄	—	134	2	no	—
200	C₂×D₄	C₈×D₆₇	1	2	yes	—
201	C₂×D₄	#11761611	1	1	yes	characteristic
202	C₂²×C₄	—	67	1	no	—
203	C₂²×C₄	—	134	1	no	—
204	C₂²×C₄	—	134	2	no	—
205	C₂²×C₄	—	134	1	no	—
206	C₂²×C₄	—	134	2	no	—
207	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
208	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
209	M₁₆	—	2	2	no	—
210	M₁₆	—	2	4	no	—
211	M₁₆	—	268	1	no	—
212	M₁₆	—	268	2	no	—
213	M₁₆	—	268	1	no	—
214	M₁₆	—	134	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	134	1	no	—
218	C₂×C₈	—	134	1	no	—
219	C₂×C₈	—	134	2	no	—
220	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
221	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
222	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
223	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
224	C₂×C₈	#11761628	1	2	yes	—
225	C₄⋊C₄	—	134	2	no	—
226	C₄⋊C₄	—	134	1	no	—
227	C₂²⋊C₄	—	67	2	no	—
228	C₂²⋊C₄	—	134	2	no	—
229	C₂²⋊C₄	—	134	2	no	—
230	C₄²	—	134	1	no	—
231	C₁₆	—	2	4	no	—
232	C₁₆	—	268	2	no	—
233	C₂³	—	536	1	no	—
234	C₂³	—	134	1	no	—
235	C₂³	—	134	2	no	—
236	C₂³	(C₂×D₆₇)⋊C₈	1	2	yes	—
237	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
238	Q₈	—	2	2	no	—
239	D₄	—	2	4	no	—
240	D₄	—	2	2	no	—
241	D₄	—	536	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	268	1	no	—
245	C₂×C₄	—	268	1	no	—
246	C₂×C₄	—	268	2	no	—
247	C₂×C₄	—	134	2	no	—
248	C₂×C₄	—	134	1	no	—
249	C₂×C₄	—	134	2	no	—
250	C₂×C₄	(C₂×D₆₇)⋊C₈	1	2	yes	—
251	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
252	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
253	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
254	C₂×C₄	#14524049	1	2	yes	—
255	C₈	—	2	2	no	—
256	C₈	—	268	1	no	—
257	C₈	—	268	1	no	—
258	C₈	#14524012	1	1	yes	characteristic
259	C₈	#14524017	1	2	yes	—
260	C₈	#14524026	1	1	yes	characteristic
261	C₂²	—	2	2	no	—
262	C₂²	—	4	1	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	268	1	no	—
265	C₂²	—	268	1	no	—
266	C₂²	—	268	1	no	—
267	C₂²	—	536	2	no	—
268	C₂²	C₂×(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
269	C₂²	C₈×C₂²⋊D₆₇	1	2	yes	—
270	C₄	—	2	2	no	—
271	C₄	—	268	2	no	—
272	C₄	—	268	1	no	—
273	C₄	#16863275	1	1	yes	characteristic
274	C₄	#16863362	1	1	yes	characteristic
275	C₄	#16863467	1	2	yes	—
276	C₂	—	2	1	no	—
277	C₂	—	2	2	no	—
278	C₂	—	4	2	no	—
279	C₂	—	536	1	no	—
280	C₂	#22560501	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22568910

C₂

yes

maximal, characteristic

#22561830

C₂

yes

maximal, characteristic, complemented

C₂×#16863177

C₂

yes

maximal, complemented

#22558296

C₂

yes

maximal, complemented

C₆₇×#3407

C₂

yes

maximal, characteristic, complemented

C₂²×#11761611

C₂²

yes

characteristic

#16863903

C₄

yes

characteristic

C₂×#14523983

C₂²

yes

—

#16863472

—

#16863467

—

#16863424

C₄

yes

characteristic

#16863363

C₄

yes

characteristic

#16863359

C₄

yes

characteristic

#16863351

C₂²

yes

characteristic

#16863178

—

#16863177

—

C₂×#131×C₆₇

C₂²

yes

characteristic

#504×C₆₇

—

C₂×M₃₂×C₆₇

—

C₂×M₃₂×C₆₇

C₂²

yes

—

C₂²×C₄×D₆₇

C₂×C₄

yes

characteristic

C₂×#11761612

C₂×C₄

yes

—

—

—

—

C₂²×C₆₇⋊C₈

C₂×C₄

yes

characteristic

#14524024

C₈

yes

—

#14524019

—

#14524014

—

#14524011

C₈

yes

—

C₄×C₂²⋊D₆₇

—

#14524002

C₂×C₄

yes

characteristic

#14523983

—

#14523983

—

C₂×C₆₇⋊C₁₆

—

Dic₆₇⋊C₈

—

C₂²⋊(C₆₇⋊C₈)

—

(C₂×D₆₇)⋊C₈

—

#14523953

—

C₈×Dic₆₇

—

C₂×#41×C₆₇

C₂×C₄

yes

characteristic

#131×C₆₇

—

#131×C₆₇

—

C₂×M₁₆×C₆₇

C₂×C₄

yes

characteristic

C₂×M₁₆×C₆₇

D₄

yes

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

M₃₂×C₆₇

—

M₃₂×C₆₇

—

M₃₂×C₆₇

D₄

yes

complemented

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

—

C₂³×D₆₇

C₂×C₈

yes

characteristic

C₂×C₂²⋊D₆₇

—

C₂²×Dic₆₇

C₂×C₈

yes

characteristic

C₂×C₄×D₆₇

—

C₂×C₄×D₆₇

—

C₂×C₄×D₆₇

—

C₂×C₄×D₆₇

—

—

—

—

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₂²⋊Dic₆₇

C₂×C₈

yes

—

(C₂×D₆₇)⋊C₄

—

C₄×Dic₆₇

—

C₆₇⋊C₁₆

—

#41×C₆₇

—

#41×C₆₇

—

C₂×Q₈×C₆₇

M₁₆

yes

characteristic

C₂×D₄×C₆₇

C₂×C₈

yes

—

C₂×D₄×C₆₇

M₁₆

yes

characteristic

C₂²×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₁₆×C₆₇

—

C₂³×C₆₇

C₂²⋊C₈

yes

—

C₂³×C₆₇

C₂²×C₈

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

100

C₂×C₄×C₆₇

—

101

C₂×C₄×C₆₇

—

102

C₂×C₄×C₆₇

C₂²⋊C₈

yes

—

103

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

104

C₂×C₄×C₆₇

C₄×D₄

yes

—

105

C₂×C₄×C₆₇

C₂×M₁₆

yes

characteristic

106

C₂×C₄×C₆₇

#131

yes

—

107

C₂²⋊D₆₇

—

108

C₂×Dic₆₇

—

109

C₂×Dic₆₇

—

110

C₂×Dic₆₇

—

111

C₄×D₆₇

—

112

C₄×D₆₇

—

113

C₄×D₆₇

—

114

C₈×C₆₇

—

115

C₈×C₆₇

#120

yes

characteristic

116

C₈×C₆₇

#121

yes

—

117

C₈×C₆₇

#123

yes

characteristic

118

C₆₇⋊C₈

—

119

C₆₇⋊C₈

—

120

C₂²×C₆₇

—

121

C₂²×C₆₇

—

122

C₂²×C₆₇

—

123

C₂²×C₆₇

C₂×C₂²⋊C₈

yes

characteristic

124

C₂²×C₆₇

C₈×D₄

yes

—

125

C₂×D₆₇

—

126

C₂×D₆₇

—

127

C₂×D₆₇

—

128

C₂×D₆₇

—

129

C₄×C₆₇

—

130

C₄×C₆₇

#386

yes

characteristic

131

C₄×C₆₇

#408

yes

characteristic

132

C₄×C₆₇

#435

yes

—

133

Dic₆₇

—

134

Dic₆₇

—

135

#12989

—

maximal, Sylow

136

C₂×C₆₇

—

137

C₂×C₆₇

—

138

C₂×C₆₇

—

139

C₂×C₆₇

#1869

yes

characteristic

140

D₆₇

—

141

#3407

D₆₇

yes

characteristic, complemented

142

#2930

—

143

#2115

—

144

C₂×#349

—

145

#1378

—

146

C₆₇

#12989

yes

characteristic, complemented, Sylow

147

C₂×#131

C₂×D₆₇

yes

characteristic

148

C₂²×M₁₆

—

149

#525

—

150

#504

—

151

C₂×M₃₂

—

152

C₂×M₃₂

—

153

C₂×M₃₂

C₂×D₆₇

yes

—

154

#436

—

134

—

155

#435

—

134

—

156

#424

—

157

#408

—

158

#407

—

159

#405

—

160

#349

—

134

—

161

#349

—

134

—

162

C₂×#41

C₄×D₆₇

yes

characteristic

163

C₂³×C₄

—

164

#131

—

165

#131

—

166

C₂×M₁₆

—

167

C₂×M₁₆

—

134

—

168

C₂×M₁₆

—

134

—

169

C₂×M₁₆

—

134

—

170

C₂×M₁₆

—

134

—

171

C₂×M₁₆

C₄×D₆₇

yes

characteristic

172

C₂×M₁₆

D₂₆₈

yes

—

173

C₂²×C₈

—

174

C₂²×C₈

C₂²⋊D₆₇

yes

—

175

C₂²×C₈

C₂²×D₆₇

yes

characteristic

176

#123

—

177

C₂²⋊Q₈

—

134

—

178

#121

—

134

—

179

#120

—

180

C₄×D₄

—

134

—

181

#117

—

182

M₃₂

—

183

M₃₂

—

184

M₃₂

—

134

—

185

M₃₂

—

134

—

186

M₃₂

C₂²⋊D₆₇

yes

complemented

187

C₂×C₁₆

—

188

C₂×C₁₆

—

189

C₂×C₁₆

—

134

—

190

C₄⋊C₈

—

134

—

191

C₂²⋊C₈

—

134

—

192

C₂²⋊C₈

—

134

—

193

#97

—

134

—

194

C₄×C₈

—

134

—

195

C₂⁴

—

196

#41

—

197

#41

—

198

C₂×Q₈

#11761611

yes

characteristic

199

C₂×D₄

—

134

—

200

C₂×D₄

C₈×D₆₇

yes

—

201

C₂×D₄

#11761611

yes

characteristic

202

C₂²×C₄

—

203

C₂²×C₄

—

134

—

204

C₂²×C₄

—

134

—

205

C₂²×C₄

—

134

—

206

C₂²×C₄

—

134

—

207

C₂²×C₄

(C₂×D₆₇)⋊C₄

yes

—

208

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

209

M₁₆

—

210

M₁₆

—

211

M₁₆

—

268

—

212

M₁₆

—

268

—

213

M₁₆

—

268

—

214

M₁₆

—

134

—

215

C₂×C₈

—

216

C₂×C₈

—

217

C₂×C₈

—

134

—

218

C₂×C₈

—

134

—

219

C₂×C₈

—

134

—

220

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

—

221

C₂×C₈

C₂×C₂²⋊D₆₇

yes

—

222

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

223

C₂×C₈

C₂×D₂₆₈

yes

characteristic

224

C₂×C₈

#11761628

yes

—

225

C₄⋊C₄

—

134

—

226

C₄⋊C₄

—

134

—

227

C₂²⋊C₄

—

228

C₂²⋊C₄

—

134

—

229

C₂²⋊C₄

—

134

—

230

C₄²

—

134

—

231

C₁₆

—

232

C₁₆

—

268

—

233

C₂³

—

536

—

234

C₂³

—

134

—

235

C₂³

—

134

—

236

C₂³

(C₂×D₆₇)⋊C₈

yes

—

237

C₂³

C₂×C₈×D₆₇

yes

characteristic

238

Q₈

—

239

D₄

—

240

D₄

—

241

D₄

—

536

—

242

C₂×C₄

—

243

C₂×C₄

—

244

C₂×C₄

—

268

—

245

C₂×C₄

—

268

—

246

C₂×C₄

—

268

—

247

C₂×C₄

—

134

—

248

C₂×C₄

—

134

—

249

C₂×C₄

—

134

—

250

C₂×C₄

(C₂×D₆₇)⋊C₈

yes

—

251

C₂×C₄

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

252

C₂×C₄

C₄×C₂²⋊D₆₇

yes

—

253

C₂×C₄

C₂×#11761611

yes

characteristic

254

C₂×C₄

#14524049

yes

—

255

C₈

—

256

C₈

—

268

—

257

C₈

—

268

—

258

C₈

#14524012

yes

characteristic

259

C₈

#14524017

yes

—

260

C₈

#14524026

yes

characteristic

261

C₂²

—

262

C₂²

—

263

C₂²

—

264

C₂²

—

268

—

265

C₂²

—

268

—

266

C₂²

—

268

—

267

C₂²

—

536

—

268

C₂²

C₂×(C₂×D₆₇)⋊C₈

yes

characteristic

269

C₂²

C₈×C₂²⋊D₆₇

yes

—

270

C₄

—

271

C₄

—

268

—

272

C₄

—

268

—

273

C₄

#16863275

yes

characteristic

274

C₄

#16863362

yes

characteristic

275

C₄

#16863467

yes

—

276

C₂

—

277

C₂

—

278

C₂

—

279

C₂

—

536

—

280

C₂

#22560501

yes

characteristic

	subgroup	G/H
center	C₄	#16863275
commutator	C₂³×C₆₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#3407	C₂
socle	C₂×C₆₇	#1869
prime radical	#16863903	C₄
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#3407	D₆₇
hypocenter	C₆₇	#12989
LCS	G ▷ C₂³×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #3407

subgroup

G/H

center

C₄

#16863275

commutator

C₂³×C₆₇

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#3407

C₂

C₂×C₆₇

C₄

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #3407

N	H ≅ G/N	type
2. #22561830	C₂	semidirect
3. C₂×#16863177	C₂	semidirect
4. #22558296	C₂	semidirect
5. C₆₇×#3407	C₂	semidirect
51. M₃₂×C₆₇	D₄	semidirect
141. #3407	D₆₇	semidirect
146. C₆₇	#12989	semidirect
186. M₃₂	C₂²⋊D₆₇	semidirect

H ≅ G/N

type

2. #22561830

C₂

semidirect

3. C₂×#16863177

C₂

semidirect

4. #22558296

C₂

semidirect

5. C₆₇×#3407

C₂

semidirect

51. M₃₂×C₆₇

D₄

semidirect

141. #3407

D₆₇

semidirect

146. C₆₇

#12989

semidirect

186. M₃₂

C₂²⋊D₆₇

semidirect

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

subgroup	index
#22568910	2
#22561830	2
C₂×#16863177	2
#22558296	2
C₆₇×#3407	2
#12989	67

subgroup

index

C₆₇×#3407

Subgroups — #30772759

Conjugacy classes of subgroups

Complements

Maximal subgroups