Group #30697802 — subgroups

Subgroups — C₆₇×#18551 #30697802

Order 17152.

918 conjugacy classes of subgroups; only the 246 normal subgroups are shown.

normal subgroups only

This group has 918 conjugacy classes of subgroups — too many to draw. Only its 246 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	C₂×Q₈²×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#3139	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#3113	C₂	1	2	yes	maximal
4	C₆₇×#1645	C₂	1	8	yes	maximal, complemented
5	C₆₇×#1558	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#1491	C₂	1	2	yes	maximal
7	Q₈²×C₆₇	—	2	2	no	—
8	Q₈²×C₆₇	—	2	4	no	—
9	Q₈²×C₆₇	C₂²	1	4	yes	—
10	#533×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈×C₆₇	—	2	2	no	—
12	C₂×C₄⋊Q₈×C₆₇	—	2	2	no	—
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₆₇	—	2	2	no	—
15	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
16	#491×C₆₇	—	2	2	no	—
17	#487×C₆₇	C₂²	1	4	yes	—
18	#483×C₆₇	—	2	2	no	—
19	#475×C₆₇	C₂²	1	4	yes	—
20	#462×C₆₇	C₂²	1	4	yes	—
21	#451×C₆₇	—	2	2	no	—
22	#441×C₆₇	—	2	4	no	—
23	#432×C₆₇	C₂²	1	2	yes	—
24	#419×C₆₇	—	2	2	no	—
25	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
26	#333×C₆₇	C₂²	1	4	yes	—
27	Q₈⋊C₈×C₆₇	C₂²	1	8	yes	—
28	C₂²×Q₈×C₆₇	—	2	2	no	—
29	C₂²×Q₈×C₆₇	D₄	1	2	yes	—
30	C₂²×Q₈×C₆₇	D₄	1	2	yes	—
31	C₂×Q₁₆×C₆₇	—	2	4	no	—
32	C₂×M₁₆×C₆₇	—	2	1	no	—
33	C₄⋊Q₈×C₆₇	—	2	8	no	—
34	C₄⋊Q₈×C₆₇	—	2	8	no	—
35	C₄⋊Q₈×C₆₇	—	2	8	no	—
36	C₄⋊Q₈×C₆₇	—	2	8	no	—
37	C₄⋊Q₈×C₆₇	—	2	2	no	—
38	C₄⋊Q₈×C₆₇	D₄	1	2	yes	—
39	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
40	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
41	#125×C₆₇	—	2	1	no	—
42	C₂²⋊Q₈×C₆₇	—	2	2	no	—
43	C₄×Q₈×C₆₇	—	2	8	no	—
44	C₄×Q₈×C₆₇	—	2	8	no	—
45	C₄×Q₈×C₆₇	—	2	2	no	—
46	C₄×Q₈×C₆₇	—	2	2	no	—
47	C₄×Q₈×C₆₇	—	2	2	no	—
48	C₄×Q₈×C₆₇	D₄	1	4	yes	—
49	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
50	#117×C₆₇	—	2	1	no	—
51	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
52	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
54	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
55	C₂×C₄²×C₆₇	—	2	2	no	—
56	C₂×C₄²×C₆₇	—	2	2	no	—
57	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
58	#106×C₆₇	—	2	4	no	—
59	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
60	#103×C₆₇	—	2	4	no	—
61	#103×C₆₇	—	2	8	no	—
62	#103×C₆₇	—	2	4	no	—
63	C₂²⋊C₈×C₆₇	—	2	2	no	—
64	#97×C₆₇	—	2	1	no	—
65	#97×C₆₇	—	2	2	no	—
66	C₄×C₈×C₆₇	C₂³	1	2	yes	—
67	C₂×Q₈×C₆₇	—	2	8	no	—
68	C₂×Q₈×C₆₇	—	2	8	no	—
69	C₂×Q₈×C₆₇	—	2	2	no	—
70	C₂×Q₈×C₆₇	—	2	2	no	—
71	C₂×Q₈×C₆₇	—	2	2	no	—
72	C₂×Q₈×C₆₇	—	4	4	no	—
73	C₂×Q₈×C₆₇	—	8	4	no	—
74	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
75	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
76	C₂²×C₄×C₆₇	—	2	2	no	—
77	C₂²×C₄×C₆₇	—	2	2	no	—
78	C₂²×C₄×C₆₇	—	2	2	no	—
79	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
81	Q₁₆×C₆₇	—	4	8	no	—
82	M₁₆×C₆₇	—	4	2	no	—
83	C₂×C₈×C₆₇	—	2	2	no	—
84	C₂×C₈×C₆₇	—	2	4	no	—
85	C₄⋊C₄×C₆₇	—	2	16	no	—
86	C₄⋊C₄×C₆₇	—	2	4	no	—
87	C₄⋊C₄×C₆₇	—	2	4	no	—
88	C₄⋊C₄×C₆₇	—	2	2	no	—
89	C₄⋊C₄×C₆₇	—	2	4	no	—
90	C₄⋊C₄×C₆₇	—	2	4	no	—
91	C₄⋊C₄×C₆₇	—	2	2	no	—
92	C₄⋊C₄×C₆₇	—	2	2	no	—
93	C₄⋊C₄×C₆₇	—	4	2	no	—
94	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
95	C₂²⋊C₄×C₆₇	—	4	1	no	—
96	C₄²×C₆₇	—	2	8	no	—
97	C₄²×C₆₇	—	2	4	no	—
98	C₄²×C₆₇	—	2	4	no	—
99	C₄²×C₆₇	—	4	1	no	—
100	C₄²×C₆₇	C₂×D₄	1	2	yes	—
101	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
103	C₂³×C₆₇	#120	1	1	yes	characteristic
104	Q₈×C₆₇	—	2	8	no	—
105	Q₈×C₆₇	—	2	4	no	—
106	Q₈×C₆₇	—	4	4	no	—
107	Q₈×C₆₇	—	4	4	no	—
108	Q₈×C₆₇	—	8	2	no	—
109	Q₈×C₆₇	—	8	8	no	—
110	Q₈×C₆₇	#137	1	8	yes	—
111	C₂×C₄×C₆₇	—	2	8	no	—
112	C₂×C₄×C₆₇	—	2	4	no	—
113	C₂×C₄×C₆₇	—	2	2	no	—
114	C₂×C₄×C₆₇	—	2	4	no	—
115	C₂×C₄×C₆₇	—	2	4	no	—
116	C₂×C₄×C₆₇	—	2	4	no	—
117	C₂×C₄×C₆₇	—	4	4	no	—
118	C₂×C₄×C₆₇	—	4	4	no	—
119	C₂×C₄×C₆₇	—	4	2	no	—
120	C₂×C₄×C₆₇	#120	1	2	yes	—
121	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
124	C₈×C₆₇	—	4	2	no	—
125	C₈×C₆₇	—	4	4	no	—
126	C₂²×C₆₇	—	2	2	no	—
127	C₂²×C₆₇	#455	1	2	yes	—
128	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
129	C₄×C₆₇	—	2	8	no	—
130	C₄×C₆₇	—	2	2	no	—
131	C₄×C₆₇	—	2	1	no	—
132	C₄×C₆₇	—	2	1	no	—
133	C₄×C₆₇	—	4	4	no	—
134	C₄×C₆₇	—	4	4	no	—
135	C₄×C₆₇	—	8	2	no	—
136	C₄×C₆₇	C₂×#137	1	4	yes	—
137	#18551	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₂×C₆₇	—	2	2	no	—
139	C₂×C₆₇	#3004	1	2	yes	—
140	C₂×C₆₇	C₂×#455	1	1	yes	characteristic
141	C₂×Q₈²	C₂×C₆₇	1	1	yes	characteristic
142	#3139	C₂×C₆₇	1	1	yes	characteristic
143	#3113	C₂×C₆₇	1	2	yes	—
144	#1645	C₂×C₆₇	1	8	yes	complemented
145	#1558	C₂×C₆₇	1	1	yes	characteristic
146	#1491	C₂×C₆₇	1	2	yes	—
147	C₆₇	#18551	1	1	yes	characteristic, direct product factor, Sylow
148	Q₈²	—	2	2	no	—
149	Q₈²	—	2	4	no	—
150	Q₈²	C₂²×C₆₇	1	4	yes	—
151	#533	C₂²×C₆₇	1	1	yes	characteristic
152	C₂×C₄⋊Q₈	—	2	2	no	—
153	C₂×C₄⋊Q₈	—	2	2	no	—
154	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
155	C₂×C₄×Q₈	—	2	2	no	—
156	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
157	#491	—	2	2	no	—
158	#487	C₂²×C₆₇	1	4	yes	—
159	#483	—	2	2	no	—
160	#475	C₂²×C₆₇	1	4	yes	—
161	#462	C₂²×C₆₇	1	4	yes	—
162	#451	—	2	2	no	—
163	#441	—	2	4	no	—
164	#432	C₂²×C₆₇	1	2	yes	—
165	#419	—	2	2	no	—
166	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
167	#333	C₂²×C₆₇	1	4	yes	—
168	Q₈⋊C₈	C₂²×C₆₇	1	8	yes	—
169	C₂²×Q₈	—	2	2	no	—
170	C₂²×Q₈	D₄×C₆₇	1	2	yes	—
171	C₂²×Q₈	D₄×C₆₇	1	2	yes	—
172	C₂×Q₁₆	—	2	4	no	—
173	C₂×M₁₆	—	2	1	no	—
174	C₄⋊Q₈	—	2	8	no	—
175	C₄⋊Q₈	—	2	8	no	—
176	C₄⋊Q₈	—	2	8	no	—
177	C₄⋊Q₈	—	2	8	no	—
178	C₄⋊Q₈	—	2	2	no	—
179	C₄⋊Q₈	D₄×C₆₇	1	2	yes	—
180	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
181	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
182	#125	—	2	1	no	—
183	C₂²⋊Q₈	—	2	2	no	—
184	C₄×Q₈	—	2	8	no	—
185	C₄×Q₈	—	2	8	no	—
186	C₄×Q₈	—	2	2	no	—
187	C₄×Q₈	—	2	2	no	—
188	C₄×Q₈	—	2	2	no	—
189	C₄×Q₈	D₄×C₆₇	1	4	yes	—
190	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
191	#117	—	2	1	no	—
192	C₂×C₄⋊C₄	—	2	4	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
196	C₂×C₄²	—	2	2	no	—
197	C₂×C₄²	—	2	2	no	—
198	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
199	#106	—	2	4	no	—
200	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
201	#103	—	2	8	no	—
202	#103	—	2	4	no	—
203	#103	—	2	4	no	—
204	C₂²⋊C₈	—	2	2	no	—
205	#97	—	2	1	no	—
206	#97	—	2	2	no	—
207	C₄×C₈	C₂³×C₆₇	1	2	yes	—
208	C₂×Q₈	—	2	8	no	—
209	C₂×Q₈	—	2	8	no	—
210	C₂×Q₈	—	2	2	no	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	4	4	no	—
214	C₂×Q₈	—	8	4	no	—
215	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
216	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
221	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
222	Q₁₆	—	4	8	no	—
223	M₁₆	—	4	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	4	no	—
226	C₄⋊C₄	—	2	16	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
236	C₂²⋊C₄	—	4	1	no	—
237	C₄²	—	2	8	no	—
238	C₄²	—	2	4	no	—
239	C₄²	—	2	4	no	—
240	C₄²	—	4	1	no	—
241	C₄²	C₂×D₄×C₆₇	1	2	yes	—
242	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
243	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
244	C₂³	#120×C₆₇	1	1	yes	characteristic
245	Q₈	—	2	8	no	—
246	Q₈	—	2	4	no	—
247	Q₈	—	4	4	no	—
248	Q₈	—	4	4	no	—
249	Q₈	—	8	2	no	—
250	Q₈	—	8	8	no	—
251	Q₈	#137×C₆₇	1	8	yes	—
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	#120×C₆₇	1	2	yes	—
262	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
263	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
264	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
265	C₈	—	4	2	no	—
266	C₈	—	4	4	no	—
267	C₂²	—	2	2	no	—
268	C₂²	#455×C₆₇	1	2	yes	—
269	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	8	no	—
273	C₄	—	2	2	no	—
274	C₄	—	4	4	no	—
275	C₄	—	4	4	no	—
276	C₄	—	8	2	no	—
277	C₄	C₂×#137×C₆₇	1	4	yes	—
278	C₂	—	2	2	no	—
279	C₂	C₆₇×#3004	1	2	yes	—
280	C₂	C₂×#455×C₆₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈²×C₆₇

C₂

yes

maximal, characteristic

C₆₇×#3139

C₂

yes

maximal, characteristic

C₆₇×#3113

C₂

yes

maximal

C₆₇×#1645

C₂

yes

maximal, complemented

C₆₇×#1558

C₂

yes

maximal, characteristic

C₆₇×#1491

C₂

yes

maximal

Q₈²×C₆₇

—

Q₈²×C₆₇

—

Q₈²×C₆₇

C₂²

yes

—

#533×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊Q₈×C₆₇

—

C₂×C₄⋊Q₈×C₆₇

—

C₂×C₄⋊Q₈×C₆₇

C₂²

yes

characteristic

C₂×C₄×Q₈×C₆₇

—

C₂×C₄×Q₈×C₆₇

C₂²

yes

—

#491×C₆₇

—

#487×C₆₇

C₂²

yes

—

#483×C₆₇

—

#475×C₆₇

C₂²

yes

—

#462×C₆₇

C₂²

yes

—

#451×C₆₇

—

#441×C₆₇

—

#432×C₆₇

C₂²

yes

—

#419×C₆₇

—

C₄×M₁₆×C₆₇

C₂²

yes

characteristic

#333×C₆₇

C₂²

yes

—

Q₈⋊C₈×C₆₇

C₂²

yes

—

C₂²×Q₈×C₆₇

—

C₂²×Q₈×C₆₇

D₄

yes

—

C₂²×Q₈×C₆₇

D₄

yes

—

C₂×Q₁₆×C₆₇

—

C₂×M₁₆×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

D₄

yes

—

C₄⋊Q₈×C₆₇

C₂³

yes

—

C₄⋊Q₈×C₆₇

C₂³

yes

—

#125×C₆₇

—

C₂²⋊Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

C₄×Q₈×C₆₇

D₄

yes

—

C₄×Q₈×C₆₇

C₂³

yes

—

#117×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

—

C₂×C₄²×C₆₇

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#106×C₆₇

—

C₄⋊C₈×C₆₇

C₂³

yes

—

#103×C₆₇

—

#103×C₆₇

—

#103×C₆₇

—

C₂²⋊C₈×C₆₇

—

#97×C₆₇

—

#97×C₆₇

—

C₄×C₈×C₆₇

C₂³

yes

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

C₂×D₄

yes

—

C₂×Q₈×C₆₇

C₂×D₄

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

Q₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₂×D₄

yes

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

100

C₄²×C₆₇

C₂×D₄

yes

—

101

C₄²×C₆₇

C₂×D₄

yes

characteristic

102

C₄²×C₆₇

C₂⁴

yes

characteristic

103

C₂³×C₆₇

#120

yes

characteristic

104

Q₈×C₆₇

—

105

Q₈×C₆₇

—

106

Q₈×C₆₇

—

107

Q₈×C₆₇

—

108

Q₈×C₆₇

—

109

Q₈×C₆₇

—

110

Q₈×C₆₇

#137

yes

—

111

C₂×C₄×C₆₇

—

112

C₂×C₄×C₆₇

—

113

C₂×C₄×C₆₇

—

114

C₂×C₄×C₆₇

—

115

C₂×C₄×C₆₇

—

116

C₂×C₄×C₆₇

—

117

C₂×C₄×C₆₇

—

118

C₂×C₄×C₆₇

—

119

C₂×C₄×C₆₇

—

120

C₂×C₄×C₆₇

#120

yes

—

121

C₂×C₄×C₆₇

#120

yes

characteristic

122

C₂×C₄×C₆₇

C₂²×D₄

yes

characteristic

123

C₂×C₄×C₆₇

C₂²×D₄

yes

—

124

C₈×C₆₇

—

125

C₈×C₆₇

—

126

C₂²×C₆₇

—

127

C₂²×C₆₇

#455

yes

—

128

C₂²×C₆₇

C₂×#120

yes

characteristic

129

C₄×C₆₇

—

130

C₄×C₆₇

—

131

C₄×C₆₇

—

132

C₄×C₆₇

—

133

C₄×C₆₇

—

134

C₄×C₆₇

—

135

C₄×C₆₇

—

136

C₄×C₆₇

C₂×#137

yes

—

137

#18551

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

138

C₂×C₆₇

—

139

C₂×C₆₇

#3004

yes

—

140

C₂×C₆₇

C₂×#455

yes

characteristic

141

C₂×Q₈²

C₂×C₆₇

yes

characteristic

142

#3139

C₂×C₆₇

yes

characteristic

143

#3113

C₂×C₆₇

yes

—

144

#1645

C₂×C₆₇

yes

complemented

145

#1558

C₂×C₆₇

yes

characteristic

146

#1491

C₂×C₆₇

yes

—

147

C₆₇

#18551

yes

characteristic, direct product factor, Sylow

148

Q₈²

—

149

Q₈²

—

150

Q₈²

C₂²×C₆₇

yes

—

151

#533

C₂²×C₆₇

yes

characteristic

152

C₂×C₄⋊Q₈

—

153

C₂×C₄⋊Q₈

—

154

C₂×C₄⋊Q₈

C₂²×C₆₇

yes

characteristic

155

C₂×C₄×Q₈

—

156

C₂×C₄×Q₈

C₂²×C₆₇

yes

—

157

#491

—

158

#487

C₂²×C₆₇

yes

—

159

#483

—

160

#475

C₂²×C₆₇

yes

—

161

#462

C₂²×C₆₇

yes

—

162

#451

—

163

#441

—

164

#432

C₂²×C₆₇

yes

—

165

#419

—

166

C₄×M₁₆

C₂²×C₆₇

yes

characteristic

167

#333

C₂²×C₆₇

yes

—

168

Q₈⋊C₈

C₂²×C₆₇

yes

—

169

C₂²×Q₈

—

170

C₂²×Q₈

D₄×C₆₇

yes

—

171

C₂²×Q₈

D₄×C₆₇

yes

—

172

C₂×Q₁₆

—

173

C₂×M₁₆

—

174

C₄⋊Q₈

—

175

C₄⋊Q₈

—

176

C₄⋊Q₈

—

177

C₄⋊Q₈

—

178

C₄⋊Q₈

—

179

C₄⋊Q₈

D₄×C₆₇

yes

—

180

C₄⋊Q₈

C₂³×C₆₇

yes

—

181

C₄⋊Q₈

C₂³×C₆₇

yes

—

182

#125

—

183

C₂²⋊Q₈

—

184

C₄×Q₈

—

185

C₄×Q₈

—

186

C₄×Q₈

—

187

C₄×Q₈

—

188

C₄×Q₈

—

189

C₄×Q₈

D₄×C₆₇

yes

—

190

C₄×Q₈

C₂³×C₆₇

yes

—

191

#117

—

192

C₂×C₄⋊C₄

—

193

C₂×C₄⋊C₄

—

194

C₂×C₄⋊C₄

—

195

C₂×C₄⋊C₄

D₄×C₆₇

yes

—

196

C₂×C₄²

—

197

C₂×C₄²

—

198

C₂×C₄²

C₂³×C₆₇

yes

characteristic

199

#106

—

200

C₄⋊C₈

C₂³×C₆₇

yes

—

201

#103

—

202

#103

—

203

#103

—

204

C₂²⋊C₈

—

205

#97

—

206

#97

—

207

C₄×C₈

C₂³×C₆₇

yes

—

208

C₂×Q₈

—

209

C₂×Q₈

—

210

C₂×Q₈

—

211

C₂×Q₈

—

212

C₂×Q₈

—

213

C₂×Q₈

—

214

C₂×Q₈

—

215

C₂×Q₈

C₂×D₄×C₆₇

yes

—

216

C₂×Q₈

C₂×D₄×C₆₇

yes

—

217

C₂²×C₄

—

218

C₂²×C₄

—

219

C₂²×C₄

—

220

C₂²×C₄

C₂×D₄×C₆₇

yes

—

221

C₂²×C₄

C₂×D₄×C₆₇

yes

characteristic

222

Q₁₆

—

223

M₁₆

—

224

C₂×C₈

—

225

C₂×C₈

—

226

C₄⋊C₄

—

227

C₄⋊C₄

—

228

C₄⋊C₄

—

229

C₄⋊C₄

—

230

C₄⋊C₄

—

231

C₄⋊C₄

—

232

C₄⋊C₄

—

233

C₄⋊C₄

—

234

C₄⋊C₄

—

235

C₄⋊C₄

C₂×D₄×C₆₇

yes

—

236

C₂²⋊C₄

—

237

C₄²

—

238

C₄²

—

239

C₄²

—

240

C₄²

—

241

C₄²

C₂×D₄×C₆₇

yes

—

242

C₄²

C₂×D₄×C₆₇

yes

characteristic

243

C₄²

C₂⁴×C₆₇

yes

characteristic

244

C₂³

#120×C₆₇

yes

characteristic

245

Q₈

—

246

Q₈

—

247

Q₈

—

248

Q₈

—

249

Q₈

—

250

Q₈

—

251

Q₈

#137×C₆₇

yes

—

252

C₂×C₄

—

253

C₂×C₄

—

254

C₂×C₄

—

255

C₂×C₄

—

256

C₂×C₄

—

257

C₂×C₄

—

258

C₂×C₄

—

259

C₂×C₄

—

260

C₂×C₄

—

261

C₂×C₄

#120×C₆₇

yes

—

262

C₂×C₄

#120×C₆₇

yes

characteristic

263

C₂×C₄

C₂²×D₄×C₆₇

yes

characteristic

264

C₂×C₄

C₂²×D₄×C₆₇

yes

—

265

C₈

—

266

C₈

—

267

C₂²

—

268

C₂²

#455×C₆₇

yes

—

269

C₂²

C₂×#120×C₆₇

yes

characteristic

270

C₄

—

271

C₄

—

272

C₄

—

273

C₄

—

274

C₄

—

275

C₄

—

276

C₄

—

277

C₄

C₂×#137×C₆₇

yes

—

278

C₂

—

279

C₂

C₆₇×#3004

yes

—

280

C₂

C₂×#455×C₆₇

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₂³×C₆₇	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₆₇

C₂×#120

commutator, Frattini, cosocle

C₄²

C₂⁴×C₆₇

C₂³×C₆₇

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

N	H ≅ G/N	type
4. C₆₇×#1645	C₂	semidirect
137. #18551	C₆₇	direct (N × H)
144. #1645	C₂×C₆₇	semidirect
147. C₆₇	#18551	direct (N × H)

H ≅ G/N

type

4. C₆₇×#1645

C₂

semidirect

137. #18551

C₆₇

direct (N × H)

144. #1645

C₂×C₆₇

semidirect

147. C₆₇

#18551

direct (N × H)

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

subgroup	index
C₂×Q₈²×C₆₇	2
C₆₇×#3139	2
C₆₇×#3113	2
C₆₇×#1645	2
C₆₇×#1558	2
C₆₇×#1491	2
#18551	67

subgroup

index

C₂×Q₈²×C₆₇

C₆₇×#3139

C₆₇×#3113

C₆₇×#1645

C₆₇×#1558

C₆₇×#1491

#18551

Subgroups — C₆₇×#18551 #30697802

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#18551 #30697802

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#18551 #30697802