Subgroups — C₂×C₆₇×#2981 #30739480

Order 17152.

2230 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#532×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2981	C₂	1	16	yes	maximal, direct product factor
3	C₂×#436×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₂×#435×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₂×C₈×D₄×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#432×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#408×C₆₇	C₂	1	4	yes	maximal, complemented
9	C₂²×#41×C₆₇	C₂²	1	2	yes	—
10	C₂²×M₁₆×C₆₇	C₂²	1	4	yes	—
11	C₂³×C₈×C₆₇	C₂²	1	4	yes	—
12	#532×C₆₇	C₄	1	8	yes	—
13	#532×C₆₇	C₂²	1	8	yes	—
14	C₂×C₄⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
15	C₂×#127×C₆₇	C₄	1	1	yes	characteristic
16	C₂×#124×C₆₇	C₄	1	2	yes	—
17	C₂×#121×C₆₇	C₄	1	2	yes	—
18	C₂×#121×C₆₇	C₄	1	2	yes	—
19	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
21	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
22	#436×C₆₇	C₂²	1	16	yes	complemented
23	#435×C₆₇	C₂²	1	32	yes	complemented
24	C₈×D₄×C₆₇	C₂²	1	16	yes	complemented
25	#432×C₆₇	C₂²	1	8	yes	complemented
26	#431×C₆₇	C₂²	1	8	yes	complemented
27	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
29	#408×C₆₇	C₂²	1	32	yes	complemented
30	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
31	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
32	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
33	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
34	C₂×#97×C₆₇	C₂²	1	2	yes	—
35	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×#41×C₆₇	—	4	8	no	—
38	C₂×#41×C₆₇	C₂×C₄	1	8	yes	—
39	C₂×#41×C₆₇	C₂³	1	8	yes	—
40	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
41	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
42	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
43	C₂²×D₄×C₆₇	C₂×C₄	1	4	yes	—
44	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
45	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
49	C₂×M₁₆×C₆₇	—	4	8	no	—
50	C₂×M₁₆×C₆₇	C₂³	1	16	yes	—
51	C₂²×C₈×C₆₇	—	4	8	no	—
52	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
53	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
54	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
55	C₂²×C₈×C₆₇	C₂³	1	16	yes	—
56	C₄⋊Q₈×C₆₇	C₂×C₄	1	8	yes	—
57	#127×C₆₇	C₂×C₄	1	8	yes	—
58	#124×C₆₇	C₂×C₄	1	16	yes	—
59	#121×C₆₇	C₂×C₄	1	16	yes	—
60	#121×C₆₇	C₂×C₄	1	16	yes	—
61	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	—
62	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	—
63	C₄×D₄×C₆₇	C₂³	1	8	yes	—
64	C₄×D₄×C₆₇	C₂³	1	8	yes	—
65	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
66	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
67	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
68	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
69	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
72	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
73	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
76	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
77	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
78	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
79	C₂²⋊C₈×C₆₇	C₂³	1	16	yes	—
80	C₂²⋊C₈×C₆₇	C₂³	1	16	yes	—
81	#97×C₆₇	C₂³	1	8	yes	—
82	C₄×C₈×C₆₇	C₂³	1	4	yes	—
83	C₄×C₈×C₆₇	C₂³	1	4	yes	—
84	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
85	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
86	#41×C₆₇	—	4	16	no	—
87	#41×C₆₇	—	4	16	no	—
88	C₂×Q₈×C₆₇	—	4	4	no	—
89	C₂×Q₈×C₆₇	C₂²×C₄	1	8	yes	—
90	C₂×D₄×C₆₇	—	2	8	no	—
91	C₂×D₄×C₆₇	—	4	4	no	—
92	C₂×D₄×C₆₇	—	4	4	no	—
93	C₂×D₄×C₆₇	—	4	8	no	—
94	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
95	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
96	C₂×D₄×C₆₇	C₂²×C₄	1	16	yes	—
97	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
98	C₂²×C₄×C₆₇	—	2	4	no	—
99	C₂²×C₄×C₆₇	—	2	8	no	—
100	C₂²×C₄×C₆₇	—	2	8	no	—
101	C₂²×C₄×C₆₇	—	2	4	no	—
102	C₂²×C₄×C₆₇	—	4	2	no	—
103	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
105	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
109	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
110	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
111	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
112	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
114	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
115	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
116	M₁₆×C₆₇	—	4	16	no	—
117	C₂×C₈×C₆₇	—	2	4	no	—
118	C₂×C₈×C₆₇	—	2	8	no	—
119	C₂×C₈×C₆₇	—	2	4	no	—
120	C₂×C₈×C₆₇	—	4	16	no	—
121	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
122	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
123	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
124	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
125	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
126	C₂²⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
127	C₂²⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
128	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
129	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
130	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
131	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
132	C₄²×C₆₇	C₂⁴	1	2	yes	—
133	C₄²×C₆₇	C₂⁴	1	2	yes	—
134	C₂³×C₆₇	—	2	2	no	—
135	C₂³×C₆₇	—	2	8	no	—
136	C₂³×C₆₇	—	2	2	no	—
137	C₂³×C₆₇	—	2	8	no	—
138	C₂³×C₆₇	—	2	2	no	—
139	C₂³×C₆₇	—	2	8	no	—
140	C₂³×C₆₇	C₂³×C₄	1	8	yes	—
141	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
142	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
143	Q₈×C₆₇	—	4	8	no	—
144	D₄×C₆₇	—	2	16	no	—
145	D₄×C₆₇	—	4	8	no	—
146	D₄×C₆₇	—	4	8	no	—
147	D₄×C₆₇	—	4	16	no	—
148	C₂×C₄×C₆₇	—	2	2	no	—
149	C₂×C₄×C₆₇	—	2	8	no	—
150	C₂×C₄×C₆₇	—	2	2	no	—
151	C₂×C₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	—	2	2	no	—
153	C₂×C₄×C₆₇	—	2	16	no	—
154	C₂×C₄×C₆₇	—	2	4	no	—
155	C₂×C₄×C₆₇	—	2	8	no	—
156	C₂×C₄×C₆₇	—	2	8	no	—
157	C₂×C₄×C₆₇	—	2	8	no	—
158	C₂×C₄×C₆₇	—	2	4	no	—
159	C₂×C₄×C₆₇	—	4	4	no	—
160	C₂×C₄×C₆₇	#131	1	4	yes	—
161	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
162	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
163	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
164	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
165	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
166	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
167	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
168	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
169	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
170	C₈×C₆₇	—	2	4	no	—
171	C₈×C₆₇	—	2	8	no	—
172	C₈×C₆₇	—	2	4	no	—
173	C₂²×C₆₇	—	2	2	no	—
174	C₂²×C₆₇	—	2	8	no	—
175	C₂²×C₆₇	—	2	2	no	—
176	C₂²×C₆₇	—	2	8	no	—
177	C₂²×C₆₇	—	2	2	no	—
178	C₂²×C₆₇	—	2	8	no	—
179	C₂²×C₆₇	—	4	4	no	—
180	C₂²×C₆₇	—	4	16	no	—
181	C₂²×C₆₇	#518	1	2	yes	—
182	C₂²×C₆₇	C₂×#131	1	2	yes	—
183	C₂²×C₆₇	#568	1	2	yes	—
184	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
185	C₄×C₆₇	—	2	2	no	—
186	C₄×C₆₇	—	2	8	no	—
187	C₄×C₆₇	—	2	2	no	—
188	C₄×C₆₇	—	2	2	no	—
189	C₄×C₆₇	—	2	2	no	—
190	C₄×C₆₇	#518	1	2	yes	—
191	C₄×C₆₇	C₂×#131	1	4	yes	—
192	C₄×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
193	C₂×#2981	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
194	C₂×C₆₇	—	4	2	no	—
195	C₂×C₆₇	—	4	8	no	—
196	C₂×C₆₇	#2981	1	4	yes	direct product factor
197	C₂×C₆₇	C₂×#518	1	1	yes	characteristic
198	C₂×C₆₇	C₂²×#131	1	1	yes	characteristic
199	C₂×C₆₇	C₂×#568	1	1	yes	characteristic
200	C₂×#532	C₂×C₆₇	1	1	yes	characteristic
201	#2981	C₂×C₆₇	1	16	yes	direct product factor
202	C₂×#436	C₂×C₆₇	1	2	yes	complemented
203	C₂×#435	C₂×C₆₇	1	4	yes	complemented
204	C₂×C₈×D₄	C₂×C₆₇	1	2	yes	complemented
205	C₂×#432	C₂×C₆₇	1	1	yes	characteristic, complemented
206	C₂×#431	C₂×C₆₇	1	1	yes	characteristic, complemented
207	C₂×#408	C₂×C₆₇	1	4	yes	complemented
208	C₆₇	C₂×#2981	1	1	yes	characteristic, direct product factor, Sylow
209	C₂²×#41	C₂²×C₆₇	1	2	yes	—
210	C₂²×M₁₆	C₂²×C₆₇	1	4	yes	—
211	C₂³×C₈	C₂²×C₆₇	1	4	yes	—
212	#532	C₄×C₆₇	1	8	yes	—
213	#532	C₂²×C₆₇	1	8	yes	—
214	C₂×C₄⋊Q₈	C₄×C₆₇	1	1	yes	characteristic
215	C₂×#127	C₄×C₆₇	1	1	yes	characteristic
216	C₂×#124	C₄×C₆₇	1	2	yes	—
217	C₂×#121	C₄×C₆₇	1	2	yes	—
218	C₂×#121	C₄×C₆₇	1	2	yes	—
219	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
220	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
221	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
222	#436	C₂²×C₆₇	1	16	yes	complemented
223	#435	C₂²×C₆₇	1	32	yes	complemented
224	C₈×D₄	C₂²×C₆₇	1	16	yes	complemented
225	#432	C₂²×C₆₇	1	8	yes	complemented
226	#431	C₂²×C₆₇	1	8	yes	complemented
227	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
228	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
229	#408	C₂²×C₆₇	1	32	yes	complemented
230	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
231	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
232	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
233	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
234	C₂×#97	C₂²×C₆₇	1	2	yes	—
235	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
236	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
237	C₂×#41	—	4	8	no	—
238	C₂×#41	C₂×C₄×C₆₇	1	8	yes	—
239	C₂×#41	C₂³×C₆₇	1	8	yes	—
240	C₂²×Q₈	C₂×C₄×C₆₇	1	2	yes	—
241	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
242	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
243	C₂²×D₄	C₂×C₄×C₆₇	1	4	yes	—
244	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
245	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
246	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
247	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
248	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
249	C₂×M₁₆	—	4	8	no	—
250	C₂×M₁₆	C₂³×C₆₇	1	16	yes	—
251	C₂²×C₈	—	4	8	no	—
252	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
253	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
254	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
255	C₂²×C₈	C₂³×C₆₇	1	16	yes	—
256	C₄⋊Q₈	C₂×C₄×C₆₇	1	8	yes	—
257	#127	C₂×C₄×C₆₇	1	8	yes	—
258	#124	C₂×C₄×C₆₇	1	16	yes	—
259	#121	C₂×C₄×C₆₇	1	16	yes	—
260	#121	C₂×C₄×C₆₇	1	16	yes	—
261	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	—
262	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	—
263	C₄×D₄	C₂³×C₆₇	1	8	yes	—
264	C₄×D₄	C₂³×C₆₇	1	8	yes	—
265	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
266	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
267	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
268	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
269	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
270	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
271	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
272	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
273	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
274	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
275	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
276	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
277	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
278	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
279	C₂²⋊C₈	C₂³×C₆₇	1	16	yes	—
280	C₂²⋊C₈	C₂³×C₆₇	1	16	yes	—
281	#97	C₂³×C₆₇	1	8	yes	—
282	C₄×C₈	C₂³×C₆₇	1	4	yes	—
283	C₄×C₈	C₂³×C₆₇	1	4	yes	—
284	C₂⁴	C₂²×C₄×C₆₇	1	4	yes	—
285	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
286	#41	—	4	16	no	—
287	#41	—	4	16	no	—
288	C₂×Q₈	—	4	4	no	—
289	C₂×Q₈	C₂²×C₄×C₆₇	1	8	yes	—
290	C₂×D₄	—	2	8	no	—
291	C₂×D₄	—	4	4	no	—
292	C₂×D₄	—	4	8	no	—
293	C₂×D₄	—	4	4	no	—
294	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
295	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
296	C₂×D₄	C₂²×C₄×C₆₇	1	16	yes	—
297	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
298	C₂²×C₄	—	2	4	no	—
299	C₂²×C₄	—	2	8	no	—
300	C₂²×C₄	—	2	8	no	—
301	C₂²×C₄	—	2	4	no	—
302	C₂²×C₄	—	4	2	no	—
303	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
304	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
305	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
306	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
307	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
308	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
309	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
310	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
311	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
312	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
313	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
314	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
315	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
316	M₁₆	—	4	16	no	—
317	C₂×C₈	—	2	4	no	—
318	C₂×C₈	—	2	8	no	—
319	C₂×C₈	—	2	4	no	—
320	C₂×C₈	—	4	16	no	—
321	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
322	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
323	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
324	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
325	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
326	C₂²⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
327	C₂²⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
328	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
329	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
330	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
331	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
332	C₄²	C₂⁴×C₆₇	1	2	yes	—
333	C₄²	C₂⁴×C₆₇	1	2	yes	—
334	C₂³	—	2	2	no	—
335	C₂³	—	2	8	no	—
336	C₂³	—	2	2	no	—
337	C₂³	—	2	8	no	—
338	C₂³	—	2	2	no	—
339	C₂³	—	2	8	no	—
340	C₂³	C₂³×C₄×C₆₇	1	8	yes	—
341	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
342	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
343	Q₈	—	4	8	no	—
344	D₄	—	2	16	no	—
345	D₄	—	4	8	no	—
346	D₄	—	4	16	no	—
347	D₄	—	4	8	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	8	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	4	no	—
354	C₂×C₄	—	2	16	no	—
355	C₂×C₄	—	2	8	no	—
356	C₂×C₄	—	2	8	no	—
357	C₂×C₄	—	2	8	no	—
358	C₂×C₄	—	2	4	no	—
359	C₂×C₄	—	4	4	no	—
360	C₂×C₄	#131×C₆₇	1	4	yes	—
361	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
362	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
363	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
364	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
365	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
366	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
367	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
368	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
369	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
370	C₈	—	2	4	no	—
371	C₈	—	2	8	no	—
372	C₈	—	2	4	no	—
373	C₂²	—	2	2	no	—
374	C₂²	—	2	8	no	—
375	C₂²	—	2	2	no	—
376	C₂²	—	2	8	no	—
377	C₂²	—	2	2	no	—
378	C₂²	—	2	8	no	—
379	C₂²	—	4	4	no	—
380	C₂²	—	4	16	no	—
381	C₂²	#518×C₆₇	1	2	yes	—
382	C₂²	C₂×#131×C₆₇	1	2	yes	—
383	C₂²	#568×C₆₇	1	2	yes	—
384	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
385	C₄	—	2	2	no	—
386	C₄	—	2	2	no	—
387	C₄	—	2	2	no	—
388	C₄	—	2	2	no	—
389	C₄	—	2	8	no	—
390	C₄	#518×C₆₇	1	2	yes	—
391	C₄	C₂×#131×C₆₇	1	4	yes	—
392	C₄	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
393	C₂	—	4	2	no	—
394	C₂	—	4	8	no	—
395	C₂	C₆₇×#2981	1	4	yes	direct product factor
396	C₂	C₂×#518×C₆₇	1	1	yes	characteristic
397	C₂	C₂²×#131×C₆₇	1	1	yes	characteristic
398	C₂	C₂×#568×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂×#532×C₆₇	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2981	C₂	direct (N × H)
3. C₂×#436×C₆₇	C₂	semidirect
4. C₂×#435×C₆₇	C₂	semidirect
5. C₂×C₈×D₄×C₆₇	C₂	semidirect
6. C₂×#432×C₆₇	C₂	semidirect
7. C₂×#431×C₆₇	C₂	semidirect
8. C₂×#408×C₆₇	C₂	semidirect
22. #436×C₆₇	C₂²	semidirect
23. #435×C₆₇	C₂²	semidirect
24. C₈×D₄×C₆₇	C₂²	semidirect
25. #432×C₆₇	C₂²	semidirect
26. #431×C₆₇	C₂²	semidirect
29. #408×C₆₇	C₂²	semidirect
193. C₂×#2981	C₆₇	direct (N × H)
196. C₂×C₆₇	#2981	direct (N × H)
201. #2981	C₂×C₆₇	direct (N × H)
202. C₂×#436	C₂×C₆₇	semidirect
203. C₂×#435	C₂×C₆₇	semidirect
204. C₂×C₈×D₄	C₂×C₆₇	semidirect
205. C₂×#432	C₂×C₆₇	semidirect
206. C₂×#431	C₂×C₆₇	semidirect
207. C₂×#408	C₂×C₆₇	semidirect
208. C₆₇	C₂×#2981	direct (N × H)
222. #436	C₂²×C₆₇	semidirect
223. #435	C₂²×C₆₇	semidirect
224. C₈×D₄	C₂²×C₆₇	semidirect
225. #432	C₂²×C₆₇	semidirect
226. #431	C₂²×C₆₇	semidirect
229. #408	C₂²×C₆₇	semidirect
395. C₂	C₆₇×#2981	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×#532×C₆₇	2
C₆₇×#2981	2
C₂×#436×C₆₇	2
C₂×#435×C₆₇	2
C₂×C₈×D₄×C₆₇	2
C₂×#432×C₆₇	2
C₂×#431×C₆₇	2
C₂×#408×C₆₇	2
C₂×#2981	67

Subgroups — C2×C67×#2981 #30739480

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2981 #30739480