Group #30773214 — subgroups

Subgroups — #30773214

Order 17152.

508 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 143 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	C₂³×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16863203	C₂	1	2	yes	maximal, complemented
3	#22560247	C₂	1	1	yes	maximal, characteristic, complemented
4	#22558304	C₂	1	2	yes	maximal, complemented
5	C₆₇×#3050	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₁₆	—	2	4	no	—
7	C₂²×C₆₇⋊C₁₆	—	2	2	no	—
8	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
9	C₂×#14523979	C₂²	1	2	yes	—
10	#16863360	—	2	1	no	—
11	#16863203	—	2	4	no	—
12	#16863161	—	2	1	no	—
13	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	#525×C₆₇	—	2	1	no	—
16	#467×C₆₇	C₄	1	2	yes	—
17	#466×C₆₇	C₄	1	2	yes	—
18	#465×C₆₇	—	2	2	no	—
19	#427×C₆₇	C₂²	1	1	yes	characteristic
20	#416×C₆₇	—	2	1	no	—
21	C₂×#11761612	—	2	2	no	—
22	C₂×C₆₇⋊C₁₆	—	2	8	no	—
23	C₂×C₆₇⋊C₁₆	—	2	2	no	—
24	C₂×C₆₇⋊C₁₆	—	2	2	no	—
25	C₂×C₆₇⋊C₁₆	D₄	1	4	yes	complemented
26	#14523979	—	2	2	no	—
27	#14523979	—	2	2	no	—
28	C₂²⋊(C₆₇⋊C₈)	—	4	2	no	—
29	C₂×#41×C₆₇	—	2	1	no	—
30	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
31	#135×C₆₇	—	2	2	no	—
32	#135×C₆₇	—	2	2	no	—
33	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂×SD₁₆×C₆₇	—	2	1	no	—
35	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₆₇	—	2	2	no	—
37	C₂²×C₈×C₆₇	—	2	2	no	—
38	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₆₇	D₄	1	2	yes	—
41	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
42	#123×C₆₇	—	4	1	no	—
43	C₂²⋊Q₈×C₆₇	—	2	2	no	—
44	#121×C₆₇	—	2	2	no	—
45	#120×C₆₇	—	4	1	no	—
46	C₄×D₄×C₆₇	—	4	2	no	—
47	#117×C₆₇	—	2	1	no	—
48	#107×C₆₇	C₂×C₄	1	2	yes	—
49	#106×C₆₇	—	2	1	no	—
50	#103×C₆₇	—	2	2	no	—
51	#102×C₆₇	—	2	2	no	—
52	#11761612	—	4	2	no	—
53	#11761612	—	4	2	no	—
54	C₂×C₆₇⋊C₈	—	4	2	no	—
55	C₆₇⋊C₁₆	—	2	4	no	—
56	C₂⁴×C₆₇	SD₁₆	1	1	yes	characteristic
57	#41×C₆₇	—	4	2	no	—
58	#41×C₆₇	—	4	2	no	—
59	C₂×Q₈×C₆₇	—	2	1	no	—
60	C₂×D₄×C₆₇	—	2	1	no	—
61	C₂×D₄×C₆₇	—	4	2	no	—
62	C₂×D₄×C₆₇	—	4	1	no	—
63	C₂²×C₄×C₆₇	—	2	2	no	—
64	C₂²×C₄×C₆₇	—	2	2	no	—
65	C₂²×C₄×C₆₇	—	4	1	no	—
66	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₆₇	D₈	1	2	yes	—
68	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
70	Q₁₆×C₆₇	—	2	2	no	—
71	SD₁₆×C₆₇	—	4	1	no	—
72	SD₁₆×C₆₇	—	4	1	no	—
73	D₈×C₆₇	—	2	2	no	—
74	C₂×C₈×C₆₇	—	2	2	no	—
75	C₂×C₈×C₆₇	—	2	4	no	—
76	C₂×C₈×C₆₇	—	2	2	no	—
77	C₂×C₈×C₆₇	—	2	2	no	—
78	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
79	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₆₇	#41	1	2	yes	—
83	C₄⋊C₄×C₆₇	—	2	2	no	—
84	C₄⋊C₄×C₆₇	—	4	2	no	—
85	C₂²⋊C₄×C₆₇	—	4	2	no	—
86	C₂²⋊C₄×C₆₇	—	4	1	no	—
87	C₂²⋊C₄×C₆₇	—	4	2	no	—
88	C₄²×C₆₇	—	4	1	no	—
89	C₂³×C₆₇	—	2	4	no	—
90	C₂³×C₆₇	—	2	2	no	—
91	C₂³×C₆₇	—	4	1	no	—
92	C₂³×C₆₇	#102	1	2	yes	—
93	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
94	Q₈×C₆₇	—	4	1	no	—
95	Q₈×C₆₇	—	4	1	no	—
96	D₄×C₆₇	—	4	1	no	—
97	D₄×C₆₇	—	4	1	no	—
98	D₄×C₆₇	—	8	4	no	—
99	D₄×C₆₇	—	8	2	no	—
100	C₂×C₄×C₆₇	—	2	2	no	—
101	C₂×C₄×C₆₇	—	2	2	no	—
102	C₂×C₄×C₆₇	—	2	2	no	—
103	C₂×C₄×C₆₇	—	2	4	no	—
104	C₂×C₄×C₆₇	—	4	2	no	—
105	C₂×C₄×C₆₇	—	4	1	no	—
106	C₂×C₄×C₆₇	—	8	2	no	—
107	C₂×C₄×C₆₇	#102	1	2	yes	—
108	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₆₇	#121	1	2	yes	—
110	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₆₇	#135	1	2	yes	—
112	C₈×C₆₇	—	2	2	no	—
113	C₈×C₆₇	C₄×D₄	1	2	yes	—
114	C₈×C₆₇	#120	1	1	yes	characteristic
115	C₈×C₆₇	#123	1	1	yes	characteristic
116	C₆₇⋊C₈	—	8	2	no	—
117	C₂²×C₆₇	—	2	8	no	—
118	C₂²×C₆₇	—	2	2	no	—
119	C₂²×C₆₇	—	2	2	no	—
120	C₂²×C₆₇	—	2	2	no	—
121	C₂²×C₆₇	—	8	2	no	—
122	C₂²×C₆₇	—	8	1	no	—
123	C₂²×C₆₇	#359	1	4	yes	complemented
124	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
125	C₂²×C₆₇	#465	1	2	yes	—
126	C₄×C₆₇	—	2	2	no	—
127	C₄×C₆₇	—	8	2	no	—
128	C₄×C₆₇	—	8	1	no	—
129	C₄×C₆₇	#386	1	1	yes	characteristic
130	C₄×C₆₇	#416	1	1	yes	characteristic
131	C₄×C₆₇	#466	1	2	yes	—
132	#13063	—	67	1	no	maximal, Sylow
133	C₂×C₆₇	—	2	4	no	—
134	C₂×C₆₇	—	2	2	no	—
135	C₂×C₆₇	—	16	1	no	—
136	C₂×C₆₇	#1893	1	1	yes	characteristic
137	C₂×C₆₇	C₂×#359	1	2	yes	—
138	C₂³×C₁₆	—	67	1	no	—
139	#3050	D₆₇	1	1	yes	characteristic
140	C₂×#359	—	67	2	no	—
141	#1810	—	67	1	no	—
142	#1382	—	67	2	no	—
143	C₆₇	#13063	1	1	yes	characteristic, complemented, Sylow
144	C₂×#135	C₂×D₆₇	1	1	yes	characteristic
145	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
146	#525	—	2	1	no	—
147	C₂²×C₁₆	—	67	2	no	—
148	C₂²×C₁₆	—	134	4	no	—
149	C₂²×C₁₆	—	134	2	no	—
150	#467	Dic₆₇	1	2	yes	—
151	#466	Dic₆₇	1	2	yes	—
152	#465	—	2	2	no	—
153	C₂×#108	—	67	2	no	—
154	#427	C₂×D₆₇	1	1	yes	characteristic
155	#416	—	2	1	no	—
156	#407	—	134	1	no	—
157	#359	—	134	4	no	—
158	#341	—	134	1	no	—
159	C₂×#41	—	2	1	no	—
160	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
161	#135	—	2	2	no	—
162	#135	—	2	2	no	—
163	C₂×Q₁₆	C₂×Dic₆₇	1	1	yes	characteristic
164	C₂×SD₁₆	—	2	1	no	—
165	C₂×D₈	C₂×Dic₆₇	1	1	yes	characteristic
166	C₂×M₁₆	—	134	2	no	—
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
170	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
171	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
172	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
173	#123	—	4	1	no	—
174	C₂²⋊Q₈	—	2	2	no	—
175	#121	—	2	2	no	—
176	#120	—	4	1	no	—
177	C₄×D₄	—	4	2	no	—
178	#117	—	2	1	no	—
179	C₂×C₁₆	—	67	4	no	—
180	C₂×C₁₆	—	134	8	no	—
181	C₂×C₁₆	—	134	2	no	—
182	C₂×C₁₆	—	134	2	no	—
183	#108	—	134	2	no	—
184	#108	—	134	2	no	—
185	#107	C₂×Dic₆₇	1	2	yes	—
186	#106	—	2	1	no	—
187	#103	—	2	2	no	—
188	#102	—	2	2	no	—
189	C₂²⋊C₈	—	268	2	no	—
190	C₂⁴	#11761616	1	1	yes	characteristic
191	#41	—	4	2	no	—
192	#41	—	4	2	no	—
193	C₂×Q₈	—	2	1	no	—
194	C₂×D₄	—	2	1	no	—
195	C₂×D₄	—	4	2	no	—
196	C₂×D₄	—	4	1	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	4	1	no	—
200	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
201	C₂²×C₄	#11761613	1	2	yes	—
202	C₂²×C₄	#11761616	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
204	Q₁₆	—	2	2	no	—
205	SD₁₆	—	4	1	no	—
206	SD₁₆	—	4	1	no	—
207	D₈	—	2	2	no	—
208	M₁₆	—	268	2	no	—
209	M₁₆	—	268	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	268	2	no	—
215	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
216	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
217	C₂×C₈	D₄×D₆₇	1	2	yes	—
218	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
219	C₂×C₈	#11761627	1	2	yes	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	4	2	no	—
222	C₂²⋊C₄	—	4	2	no	—
223	C₂²⋊C₄	—	4	1	no	—
224	C₂²⋊C₄	—	4	2	no	—
225	C₄²	—	4	1	no	—
226	C₁₆	—	134	4	no	—
227	C₂³	—	2	2	no	—
228	C₂³	—	2	4	no	—
229	C₂³	—	4	1	no	—
230	C₂³	#14523964	1	2	yes	—
231	C₂³	C₂×#11761616	1	1	yes	characteristic
232	Q₈	—	4	1	no	—
233	Q₈	—	4	1	no	—
234	D₄	—	4	1	no	—
235	D₄	—	4	1	no	—
236	D₄	—	8	2	no	—
237	D₄	—	8	4	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	8	2	no	—
245	C₂×C₄	#14523964	1	2	yes	—
246	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
247	C₂×C₄	#14524015	1	2	yes	—
248	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
249	C₂×C₄	#14524063	1	2	yes	—
250	C₈	—	2	2	no	—
251	C₈	—	536	2	no	—
252	C₈	D₄×Dic₆₇	1	2	yes	—
253	C₈	#14524011	1	1	yes	characteristic
254	C₈	#14524024	1	1	yes	characteristic
255	C₂²	—	2	8	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	8	1	no	—
260	C₂²	—	8	2	no	—
261	C₂²	#16863203	1	4	yes	complemented
262	C₂²	C₂×#14523964	1	1	yes	characteristic
263	C₂²	#16863623	1	2	yes	—
264	C₄	—	2	2	no	—
265	C₄	—	8	1	no	—
266	C₄	—	8	2	no	—
267	C₄	#16863276	1	1	yes	characteristic
268	C₄	#16863392	1	1	yes	characteristic
269	C₄	#16863630	1	2	yes	—
270	C₂	—	2	2	no	—
271	C₂	—	2	4	no	—
272	C₂	—	16	1	no	—
273	C₂	#22560638	1	1	yes	characteristic
274	C₂	C₂×#16863203	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₆₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₂×#16863203

C₂

yes

maximal, complemented

#22560247

C₂

yes

maximal, characteristic, complemented

#22558304

C₂

yes

maximal, complemented

C₆₇×#3050

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₁₆

—

C₂²×C₆₇⋊C₁₆

—

C₂²×C₆₇⋊C₁₆

C₂²

yes

—

C₂×#14523979

C₂²

yes

—

#16863360

—

#16863203

—

#16863161

—

C₂×#135×C₆₇

C₂²

yes

characteristic

C₂³×C₈×C₆₇

C₂²

yes

characteristic

#525×C₆₇

—

#467×C₆₇

C₄

yes

—

#466×C₆₇

C₄

yes

—

#465×C₆₇

—

#427×C₆₇

C₂²

yes

characteristic

#416×C₆₇

—

C₂×#11761612

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

D₄

yes

complemented

#14523979

—

#14523979

—

C₂²⋊(C₆₇⋊C₈)

—

C₂×#41×C₆₇

—

C₂³×C₄×C₆₇

D₄

yes

characteristic

#135×C₆₇

—

#135×C₆₇

—

C₂×Q₁₆×C₆₇

C₂×C₄

yes

characteristic

C₂×SD₁₆×C₆₇

—

C₂×D₈×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₇

—

C₂²×C₈×C₆₇

—

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

D₄

yes

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

#123×C₆₇

—

C₂²⋊Q₈×C₆₇

—

#121×C₆₇

—

#120×C₆₇

—

C₄×D₄×C₆₇

—

#117×C₆₇

—

#107×C₆₇

C₂×C₄

yes

—

#106×C₆₇

—

#103×C₆₇

—

#102×C₆₇

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₆₇⋊C₁₆

—

C₂⁴×C₆₇

SD₁₆

yes

characteristic

#41×C₆₇

—

#41×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₇

D₈

yes

—

C₂²×C₄×C₆₇

SD₁₆

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

Q₁₆×C₆₇

—

SD₁₆×C₆₇

—

SD₁₆×C₆₇

—

D₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

#102

yes

—

C₂³×C₆₇

C₂×SD₁₆

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

100

C₂×C₄×C₆₇

—

101

C₂×C₄×C₆₇

—

102

C₂×C₄×C₆₇

—

103

C₂×C₄×C₆₇

—

104

C₂×C₄×C₆₇

—

105

C₂×C₄×C₆₇

—

106

C₂×C₄×C₆₇

—

107

C₂×C₄×C₆₇

#102

yes

—

108

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

109

C₂×C₄×C₆₇

#121

yes

—

110

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

111

C₂×C₄×C₆₇

#135

yes

—

112

C₈×C₆₇

—

113

C₈×C₆₇

C₄×D₄

yes

—

114

C₈×C₆₇

#120

yes

characteristic

115

C₈×C₆₇

#123

yes

characteristic

116

C₆₇⋊C₈

—

117

C₂²×C₆₇

—

118

C₂²×C₆₇

—

119

C₂²×C₆₇

—

120

C₂²×C₆₇

—

121

C₂²×C₆₇

—

122

C₂²×C₆₇

—

123

C₂²×C₆₇

#359

yes

complemented

124

C₂²×C₆₇

C₂×#102

yes

characteristic

125

C₂²×C₆₇

#465

yes

—

126

C₄×C₆₇

—

127

C₄×C₆₇

—

128

C₄×C₆₇

—

129

C₄×C₆₇

#386

yes

characteristic

130

C₄×C₆₇

#416

yes

characteristic

131

C₄×C₆₇

#466

yes

—

132

#13063

—

maximal, Sylow

133

C₂×C₆₇

—

134

C₂×C₆₇

—

135

C₂×C₆₇

—

136

C₂×C₆₇

#1893

yes

characteristic

137

C₂×C₆₇

C₂×#359

yes

—

138

C₂³×C₁₆

—

139

#3050

D₆₇

yes

characteristic

140

C₂×#359

—

141

#1810

—

142

#1382

—

143

C₆₇

#13063

yes

characteristic, complemented, Sylow

144

C₂×#135

C₂×D₆₇

yes

characteristic

145

C₂³×C₈

C₂×D₆₇

yes

characteristic

146

#525

—

147

C₂²×C₁₆

—

148

C₂²×C₁₆

—

134

—

149

C₂²×C₁₆

—

134

—

150

#467

Dic₆₇

yes

—

151

#466

Dic₆₇

yes

—

152

#465

—

153

C₂×#108

—

154

#427

C₂×D₆₇

yes

characteristic

155

#416

—

156

#407

—

134

—

157

#359

—

134

—

158

#341

—

134

—

159

C₂×#41

—

160

C₂³×C₄

C₂²⋊D₆₇

yes

characteristic

161

#135

—

162

#135

—

163

C₂×Q₁₆

C₂×Dic₆₇

yes

characteristic

164

C₂×SD₁₆

—

165

C₂×D₈

C₂×Dic₆₇

yes

characteristic

166

C₂×M₁₆

—

134

—

167

C₂²×C₈

—

168

C₂²×C₈

—

169

C₂²×C₈

C₂×Dic₆₇

yes

—

170

C₂²×C₈

C₂²⋊D₆₇

yes

characteristic

171

C₂²×C₈

C₂²⋊D₆₇

yes

—

172

C₂²×C₈

C₂²×D₆₇

yes

characteristic

173

#123

—

174

C₂²⋊Q₈

—

175

#121

—

176

#120

—

177

C₄×D₄

—

178

#117

—

179

C₂×C₁₆

—

180

C₂×C₁₆

—

134

—

181

C₂×C₁₆

—

134

—

182

C₂×C₁₆

—

134

—

183

#108

—

134

—

184

#108

—

134

—

185

#107

C₂×Dic₆₇

yes

—

186

#106

—

187

#103

—

188

#102

—

189

C₂²⋊C₈

—

268

—

190

C₂⁴

#11761616

yes

characteristic

191

#41

—

192

#41

—

193

C₂×Q₈

—

194

C₂×D₄

—

195

C₂×D₄

—

196

C₂×D₄

—

197

C₂²×C₄

—

198

C₂²×C₄

—

199

C₂²×C₄

—

200

C₂²×C₄

C₂²⋊Dic₆₇

yes

—

201

C₂²×C₄

#11761613

yes

—

202

C₂²×C₄

#11761616

yes

characteristic

203

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

204

Q₁₆

—

205

SD₁₆

—

206

SD₁₆

—

207

D₈

—

208

M₁₆

—

268

—

209

M₁₆

—

268

—

210

C₂×C₈

—

211

C₂×C₈

—

212

C₂×C₈

—

213

C₂×C₈

—

214

C₂×C₈

—

268

—

215

C₂×C₈

C₂²⋊Dic₆₇

yes

—

216

C₂×C₈

C₂²×Dic₆₇

yes

characteristic

217

C₂×C₈

D₄×D₆₇

yes

—

218

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

219

C₂×C₈

#11761627

yes

—

220

C₄⋊C₄

—

221

C₄⋊C₄

—

222

C₂²⋊C₄

—

223

C₂²⋊C₄

—

224

C₂²⋊C₄

—

225

C₄²

—

226

C₁₆

—

134

—

227

C₂³

—

228

C₂³

—

229

C₂³

—

230

C₂³

#14523964

yes

—

231

C₂³

C₂×#11761616

yes

characteristic

232

Q₈

—

233

Q₈

—

234

D₄

—

235

D₄

—

236

D₄

—

237

D₄

—

238

C₂×C₄

—

239

C₂×C₄

—

240

C₂×C₄

—

241

C₂×C₄

—

242

C₂×C₄

—

243

C₂×C₄

—

244

C₂×C₄

—

245

C₂×C₄

#14523964

yes

—

246

C₂×C₄

C₂×C₂²⋊Dic₆₇

yes

characteristic

247

C₂×C₄

#14524015

yes

—

248

C₂×C₄

C₂×#11761613

yes

characteristic

249

C₂×C₄

#14524063

yes

—

250

C₈

—

251

C₈

—

536

—

252

C₈

D₄×Dic₆₇

yes

—

253

C₈

#14524011

yes

characteristic

254

C₈

#14524024

yes

characteristic

255

C₂²

—

256

C₂²

—

257

C₂²

—

258

C₂²

—

259

C₂²

—

260

C₂²

—

261

C₂²

#16863203

yes

complemented

262

C₂²

C₂×#14523964

yes

characteristic

263

C₂²

#16863623

yes

—

264

C₄

—

265

C₄

—

266

C₄

—

267

C₄

#16863276

yes

characteristic

268

C₄

#16863392

yes

characteristic

269

C₄

#16863630

yes

—

270

C₂

—

271

C₂

—

272

C₂

—

273

C₂

#22560638

yes

characteristic

274

C₂

C₂×#16863203

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×#11761613
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting, prime radical	C₆₇×#3050	C₂
socle	C₂²×C₆₇	C₂×#102
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#3050	D₆₇
hypocenter	C₆₇	#13063
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #3050

subgroup

G/H

center

C₂×C₄

C₂×#11761613

commutator

C₂×C₈×C₆₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting, prime radical

C₆₇×#3050

C₂

socle

C₂²×C₆₇

C₂×#102

cosocle

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #3050

N	H ≅ G/N	type
1. C₂³×C₆₇⋊C₁₆	C₂	semidirect
2. C₂×#16863203	C₂	semidirect
3. #22560247	C₂	semidirect
4. #22558304	C₂	semidirect
25. C₂×C₆₇⋊C₁₆	D₄	semidirect
123. C₂²×C₆₇	#359	semidirect
143. C₆₇	#13063	semidirect
261. C₂²	#16863203	semidirect

H ≅ G/N

type

1. C₂³×C₆₇⋊C₁₆

C₂

semidirect

2. C₂×#16863203

C₂

semidirect

3. #22560247

C₂

semidirect

4. #22558304

C₂

semidirect

25. C₂×C₆₇⋊C₁₆

D₄

semidirect

123. C₂²×C₆₇

#359

semidirect

143. C₆₇

#13063

semidirect

261. C₂²

#16863203

semidirect

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

subgroup	index
C₂³×C₆₇⋊C₁₆	2
C₂×#16863203	2
#22560247	2
#22558304	2
C₆₇×#3050	2
#13063	67

subgroup

index

C₂³×C₆₇⋊C₁₆

C₆₇×#3050

Subgroups — #30773214

Conjugacy classes of subgroups

Complements

Maximal subgroups